SystemsOfODEs - Maple Help

ODE Steps for Systems of ODEs

Overview

 • This help page gives a few examples of using the command ODESteps to solve systems of ordinary differential equations.
 • See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right):$
 > $\mathrm{high_order_ode1}≔\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3\mathrm{diff}\left(y\left(x\right),x,x\right)+4\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{high_order_ode1}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{ODESteps}\left(\mathrm{high_order_ode1}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& {\text{Highest derivative means the order of the ODE is}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{3}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{▫}& {}& \text{Convert linear ODE into a system of first order ODEs}\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{1}}{}\left({x}\right)\\ {}& {}& {{y}}_{{1}}{}\left({x}\right){=}{y}{}\left({x}\right)\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{2}}{}\left({x}\right)\\ {}& {}& {{y}}_{{2}}{}\left({x}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{3}}{}\left({x}\right)\\ {}& {}& {{y}}_{{3}}{}\left({x}\right){=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& {\text{Isolate for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{using original ODE}}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({x}\right){=}{-}{3}{}{{y}}_{{3}}{}\left({x}\right){-}{4}{}{{y}}_{{2}}{}\left({x}\right){-}{2}{}{{y}}_{{1}}{}\left({x}\right)\\ {}& {}& \text{Convert linear ODE into a system of first order ODEs}\\ {}& {}& \left[{{y}}_{{2}}{}\left({x}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){,}{{y}}_{{3}}{}\left({x}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({x}\right){=}{-}{3}{}{{y}}_{{3}}{}\left({x}\right){-}{4}{}{{y}}_{{2}}{}\left({x}\right){-}{2}{}{{y}}_{{1}}{}\left({x}\right)\right]\\ \text{•}& {}& \text{Define vector}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{c}{{y}}_{{3}}{}\left({x}\right)\\ {{y}}_{{1}}{}\left({x}\right)\\ {{y}}_{{2}}{}\left({x}\right)\end{array}\right]\\ \text{•}& {}& \text{System to solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}{A}{·}\stackrel{{\to }}{{y}}{}\left({x}\right)\\ \text{•}& {}& {\text{To solve the system find eigenvalues and eigenvectors of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {A}{=}\left[\begin{array}{ccc}{-3}& {-2}& {-4}\\ {0}& {0}& {1}\\ {1}& {0}& {0}\end{array}\right]\\ \text{•}& {}& \text{Eigenpairs of A}\\ {}& {}& \left[\left[{-1}{,}\left[\begin{array}{c}{-1}\\ {-1}\\ {1}\end{array}\right]\right]{,}\left[{-1}{+}{I}{,}\left[\begin{array}{c}{-1}{+}{I}\\ {-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\\ {1}\end{array}\right]\right]{,}\left[{-1}{-}{I}{,}\left[\begin{array}{c}{-1}{-}{I}\\ {-}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\\ {1}\end{array}\right]\right]\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{-1}{,}\left[\begin{array}{c}{-1}\\ {-1}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& \text{Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\\ {}& {}& \left[{-1}{+}{I}{,}\left[\begin{array}{c}{-1}{+}{I}\\ {-}\frac{{1}}{{2}}{-}\frac{{I}}{{2}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution from eigenpair}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Use Euler identity to write solution in terms of sin and cos}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Simplify expression}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Both real and imaginary parts are solutions to the homogeneous system}\\ {}& {}& \left[{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){=}\left[{}\right]{,}{\stackrel{{\to }}{{y}}}_{{3}}{}\left({x}\right){=}\left[{}\right]\right]\\ \text{•}& {}& \text{General solution to the system of ODEs}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){+}{\mathrm{_C3}}{}{\stackrel{{\to }}{{y}}}_{{3}}{}\left({x}\right)\\ \text{•}& {}& \text{Substitute solutions into the general solution}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[{}\right]{+}\left[{}\right]{+}\left[{}\right]\\ \text{•}& {}& \text{First component of the vector is the solution to the ODE}\\ {}& {}& {y}{}\left({x}\right){=}{-}{{ⅇ}}^{{-}{x}}{}\left(\left({\mathrm{_C2}}{+}{\mathrm{_C3}}\right){}{\mathrm{sin}}{}\left({x}\right){+}\left({\mathrm{_C2}}{-}{\mathrm{_C3}}\right){}{\mathrm{cos}}{}\left({x}\right){+}{\mathrm{_C1}}\right)\end{array}$ (2)
 > $\mathrm{macro}\left(Y=\stackrel{\to }{y}\right):$
 > $\mathrm{sys2}≔\mathrm{diff}\left(Y\left(x\right),x\right)=\mathrm{Matrix}\left(\left[\left[7,1\right],\left[-4,3\right]\right]\right)·Y\left(x\right)$
 ${\mathrm{sys2}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]{·}\stackrel{{\to }}{{y}}{}\left({x}\right)$ (3)
 > $\mathrm{ODESteps}\left(\mathrm{sys2}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]{·}\stackrel{{\to }}{{y}}{}\left({x}\right)\\ \text{•}& {}& \text{System to solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}{A}{·}\stackrel{{\to }}{{y}}{}\left({x}\right)\\ \text{•}& {}& {\text{To solve the system find eigenvalues and eigenvectors of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {A}{=}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]\\ \text{•}& {}& \text{Eigenpairs of A}\\ {}& {}& \left[\left[{5}{,}\left[\begin{array}{c}{-}\frac{{1}}{{2}}\\ {1}\end{array}\right]\right]{,}\left[{5}{,}\left[\begin{array}{c}{0}\\ {0}\end{array}\right]\right]\right]\\ \text{•}& {}& \text{Consider eigenpair, with eigenvalue of algebraic multiplicity 2}\\ {}& {}& \left[{5}{,}\left[\begin{array}{c}{-}\frac{{1}}{{2}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& {\text{First solution from eigenvalue}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{5}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& {\text{Form of the 2nd homogeneous solution where}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{p}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is to be solved for,}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\mathrm{\lambda }}{=}{5}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is the eigenvalue, and}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is the eigenvector}}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){=}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\left({x}{}\stackrel{{\to }}{{v}}{+}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}\right)\\ \text{•}& {}& {\text{Note that the}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{multiplying}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{makes this solution linearly independent to the 1st solution obtained from}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\mathrm{\lambda }}{=}{5}\\ \text{•}& {}& {\text{substitute}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{into the system}}\\ {}& {}& {\mathrm{\lambda }}{}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\left({x}{}\stackrel{{\to }}{{v}}{+}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}\right){+}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\stackrel{{\to }}{{v}}{=}\left({{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}{A}\right){·}\left({x}{}\stackrel{{\to }}{{v}}{+}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}\right)\\ \text{•}& {}& {\text{Use the fact that}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is an eigenvector of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {\mathrm{\lambda }}{}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\left({x}{}\stackrel{{\to }}{{v}}{+}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}\right){+}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\stackrel{{\to }}{{v}}{=}{{ⅇ}}^{{\mathrm{\lambda }}{}{x}}{}\left({\mathrm{\lambda }}{}\stackrel{{\to }}{{v}}{+}{\mathrm{\lambda }}{}{x}{}\stackrel{{\to }}{{v}}{+}{A}{·}\stackrel{{\to }}{{p}}\right)\\ \text{•}& {}& \text{Simplify equation}\\ {}& {}& {\mathrm{\lambda }}{}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}{=}{A}{·}\stackrel{{\to }}{{p}}\\ \text{•}& {}& {\text{Make use of the identity matrix}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{I}\\ {}& {}& \left({\mathrm{\lambda }}{}{I}\right){·}\stackrel{{\to }}{{p}}{+}\stackrel{{\to }}{{v}}{=}{A}{·}\stackrel{{\to }}{{p}}\\ \text{•}& {}& {\text{Condition}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{p}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{must meet for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{to be a solution to the system}}\\ {}& {}& \left({-}{\mathrm{\lambda }}{}{I}{+}{A}\right){·}\stackrel{{\to }}{{p}}{=}\stackrel{{\to }}{{v}}\\ \text{•}& {}& {\text{Choose}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{p}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{to use in the second solution to the system from eigenvalue}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{5}\\ {}& {}& \left(\left[{}\right]\right){·}\stackrel{{\to }}{{p}}{=}\left[\begin{array}{c}{-}\frac{{1}}{{2}}\\ {1}\end{array}\right]\\ \text{•}& {}& {\text{Choice of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{p}}\\ {}& {}& \stackrel{{\to }}{{p}}{=}\left[\begin{array}{c}{-}\frac{{1}}{{4}}\\ {0}\end{array}\right]\\ \text{•}& {}& {\text{second solution from eigenvalue}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{5}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& \text{General solution to the system of ODEs}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right)\\ \text{•}& {}& \text{Substitute solutions into the general solution}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[{}\right]{+}\left[{}\right]\end{array}$ (4)
 > $\mathrm{sys3}≔⟨\mathrm{diff}\left(y\left[1\right]\left(x\right),x\right),\mathrm{diff}\left(y\left[2\right]\left(x\right),x\right)⟩=\mathrm{Matrix}\left(\left[\left[1,2\right],\left[3,2\right]\right]\right)·⟨y\left[1\right]\left(x\right),y\left[2\right]\left(x\right)⟩+⟨1,\mathrm{exp}\left(x\right)⟩$
 ${\mathrm{sys3}}{≔}\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{1}\\ {3}{}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\end{array}\right]$ (5)
 > $\mathrm{ODESteps}\left(\mathrm{sys3}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{1}\\ {3}{}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& \text{Define vector}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right)\\ {{y}}_{{2}}{}\left({x}\right)\end{array}\right]\\ \text{•}& {}& \text{System to solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{cc}{1}& {2}\\ {3}& {2}\end{array}\right]{·}\stackrel{{\to }}{{y}}{}\left({x}\right){+}\left[\begin{array}{c}{1}\\ {{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& \text{Define forcing function}\\ {}& {}& \stackrel{{\to }}{{f}}{}\left({x}\right){=}\left[\begin{array}{c}{1}\\ {{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& {\text{To solve the system find eigenvalues and eigenvectors of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {A}{=}\left[\begin{array}{cc}{1}& {2}\\ {3}& {2}\end{array}\right]\\ \text{•}& {}& \text{Eigenpairs of A}\\ {}& {}& \left[\left[{4}{,}\left[\begin{array}{c}\frac{{2}}{{3}}\\ {1}\end{array}\right]\right]{,}\left[{-1}{,}\left[\begin{array}{c}{-1}\\ {1}\end{array}\right]\right]\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{4}{,}\left[\begin{array}{c}\frac{{2}}{{3}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{-1}{,}\left[\begin{array}{c}{-1}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& {\text{General solution to the system of ODEs where}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{p}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is a particular solution to the system}}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){+}{\stackrel{{\to }}{{y}}}_{{p}}{}\left({x}\right)\\ \text{▫}& {}& \text{Fundamental matrix}\\ {}& \text{◦}& {\text{The fundamental matrix is defined as the matrix with columns that are the solutions to the homogeneous system with initial condition being the basis vector in the standard basis at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right)\\ {}& \text{◦}& {\text{Compute the solution to the homogeneous system for each basis vector of the standard basis being the inital condition at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}{A}{·}\stackrel{{\to }}{{y}}{}\left({x}\right)\\ {}& \text{◦}& {\text{Let}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{B}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{be the matrix with solutions to the homogeneous system as columns evaluated at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& {B}{=}\left[\begin{array}{cc}\frac{{2}}{{3}}& {-1}\\ {1}& {1}\end{array}\right]\\ {}& \text{◦}& {\text{For each basis vector we need to solve the system of linear equations for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{which contains the values to multiply each homogeneous solution by to get the basis vector as the intital condition}}\\ {}& {}& {B}{·}{\stackrel{{\to }}{{C}}}_{{j}}{=}{\stackrel{{ˆ}}{{e}}}_{{j}}\\ {}& \text{◦}& \text{Equation which must be executed for each basis vector}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{j}}{=}\left[{}\right]{·}{\stackrel{{ˆ}}{{e}}}_{{j}}\\ {}& \text{◦}& {\text{Compute}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{for j =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{1}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{1}}{=}\left[\begin{array}{c}\frac{{3}}{{5}}\\ {-}\frac{{3}}{{5}}\end{array}\right]\\ {}& \text{◦}& \text{1st column of the fundamental matrix}\\ {}& {}& \left[{}\right]{+}\left[{}\right]{=}\left[\begin{array}{c}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ {-}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ {}& \text{◦}& {\text{Compute}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{for j =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{2}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{2}}{=}\left[\begin{array}{c}\frac{{3}}{{5}}\\ \frac{{2}}{{5}}\end{array}\right]\\ {}& \text{◦}& \text{2nd column of the fundamental matrix}\\ {}& {}& \left[{}\right]{+}\left[{}\right]{=}\left[\begin{array}{c}{-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ \frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ {}& {}& \text{Fundamental matrix}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right){=}\left[\begin{array}{cc}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}& {-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ {-}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}& \frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ \text{▫}& {}& \text{Find a particular solution to the system of ODEs using variation of paramaters}\\ {}& \text{◦}& {\text{Let the particular solution be the fundamental matrix multiplied by}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{and solve for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& \text{◦}& \text{Take the derivative of the particular solution}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Phi }}{}\left({x}\right)\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right)\\ {}& \text{◦}& \text{Substitute particular solution and it's derivative into the system of ODEs}\\ {}& {}& \left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Phi }}{}\left({x}\right)\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}{A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{The fundamental matrix has columns that are solutions to the homogeneous system so it's derivative follows that of the homogeneous system}\\ {}& {}& {A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}{A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{Cancel like terms}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{Multiply by the inverse of the fundamental matrix}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right){=}\left[{}\right]{·}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& {\text{Integrate to solve for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& {}& \stackrel{{\to }}{{v}}{}\left({x}\right){=}{{\int }}_{{0}}^{{x}}\left[{}\right]{·}\stackrel{{\to }}{{f}}{}\left({s}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{s}\\ {}& \text{◦}& {\text{Plug}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{into the equation for the particular solution}}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}{\mathrm{\Phi }}{}\left({x}\right){·}\left({{\int }}_{{0}}^{{x}}\left[{}\right]{·}\stackrel{{\to }}{{f}}{}\left({s}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{s}\right)\\ {}& \text{◦}& \text{Plug in the fundamental matrix and the forcing function and compute}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}\left[\begin{array}{c}\frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{30}}{-}\frac{{{ⅇ}}^{{x}}}{{3}}{+}\frac{{1}}{{2}}{-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\\ \frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{20}}{-}\frac{{3}}{{4}}{+}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\end{array}\right]\\ {}& {}& \text{Find a particular solution to the system of ODEs using variation of paramaters}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}\left[\begin{array}{c}\frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{30}}{-}\frac{{{ⅇ}}^{{x}}}{{3}}{+}\frac{{1}}{{2}}{-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\\ \frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{20}}{-}\frac{{3}}{{4}}{+}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\end{array}\right]\\ \text{•}& {}& \text{Plug particular solution back into general solution}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){+}\left[\begin{array}{c}\frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{30}}{-}\frac{{{ⅇ}}^{{x}}}{{3}}{+}\frac{{1}}{{2}}{-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\\ \frac{{7}{}{{ⅇ}}^{{4}{}{x}}}{{20}}{-}\frac{{3}}{{4}}{+}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}\end{array}\right]\\ \text{•}& {}& \text{Solution to the system of ODEs}\\ {}& {}& \left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right)\\ {{y}}_{{2}}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}\frac{\left({-}{30}{}{\mathrm{_C2}}{-}{12}\right){}{{ⅇ}}^{{-}{x}}}{{30}}{+}\frac{\left({20}{}{\mathrm{_C1}}{+}{7}\right){}{{ⅇ}}^{{4}{}{x}}}{{30}}{-}\frac{{{ⅇ}}^{{x}}}{{3}}{+}\frac{{1}}{{2}}\\ \frac{\left({20}{}{\mathrm{_C2}}{+}{8}\right){}{{ⅇ}}^{{-}{x}}}{{20}}{-}\frac{{3}}{{4}}{+}\frac{\left({20}{}{\mathrm{_C1}}{+}{7}\right){}{{ⅇ}}^{{4}{}{x}}}{{20}}\end{array}\right]\end{array}$ (6)
 > $\mathrm{sys4}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right),x\right)=y\left[1\right]\left(x\right)+2y\left[2\right]\left(x\right),\mathrm{diff}\left(y\left[2\right]\left(x\right),x\right)=3y\left[1\right]\left(x\right)+2y\left[2\right]\left(x\right)+\mathrm{exp}\left(x\right)\right\}$
 ${\mathrm{sys4}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{3}{}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\right\}$ (7)
 > $\mathrm{ODESteps}\left(\mathrm{sys4}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{3}{}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\right\}\\ \text{•}& {}& \text{Define vector}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right)\\ {{y}}_{{2}}{}\left({x}\right)\end{array}\right]\\ \text{•}& {}& \text{Convert system into a vector equation}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[{}\right]{+}\left[\begin{array}{c}{0}\\ {{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& \text{System to solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}\left[\begin{array}{cc}{1}& {2}\\ {3}& {2}\end{array}\right]{·}\stackrel{{\to }}{{y}}{}\left({x}\right){+}\left[\begin{array}{c}{0}\\ {{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& \text{Define forcing function}\\ {}& {}& \stackrel{{\to }}{{f}}{}\left({x}\right){=}\left[\begin{array}{c}{0}\\ {{ⅇ}}^{{x}}\end{array}\right]\\ \text{•}& {}& {\text{To solve the system find eigenvalues and eigenvectors of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {A}{=}\left[\begin{array}{cc}{1}& {2}\\ {3}& {2}\end{array}\right]\\ \text{•}& {}& \text{Eigenpairs of A}\\ {}& {}& \left[\left[{-1}{,}\left[\begin{array}{c}{-1}\\ {1}\end{array}\right]\right]{,}\left[{4}{,}\left[\begin{array}{c}\frac{{2}}{{3}}\\ {1}\end{array}\right]\right]\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{-1}{,}\left[\begin{array}{c}{-1}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{4}{,}\left[\begin{array}{c}\frac{{2}}{{3}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){=}\left[{}\right]\\ \text{•}& {}& {\text{General solution to the system of ODEs where}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{p}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is a particular solution to the system}}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({x}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({x}\right){+}{\stackrel{{\to }}{{y}}}_{{p}}{}\left({x}\right)\\ \text{▫}& {}& \text{Fundamental matrix}\\ {}& \text{◦}& {\text{The fundamental matrix is defined as the matrix with columns that are the solutions to the homogeneous system with initial condition being the basis vector in the standard basis at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right)\\ {}& \text{◦}& {\text{Compute the solution to the homogeneous system for each basis vector of the standard basis being the inital condition at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({x}\right){=}{A}{·}\stackrel{{\to }}{{y}}{}\left({x}\right)\\ {}& \text{◦}& {\text{Let}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{B}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{be the matrix with solutions to the homogeneous system as columns evaluated at}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{=}{0}\\ {}& {}& {B}{=}\left[\begin{array}{cc}{-1}& \frac{{2}}{{3}}\\ {1}& {1}\end{array}\right]\\ {}& \text{◦}& {\text{For each basis vector we need to solve the system of linear equations for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{which contains the values to multiply each homogeneous solution by to get the basis vector as the intital condition}}\\ {}& {}& {B}{·}{\stackrel{{\to }}{{C}}}_{{j}}{=}{\stackrel{{ˆ}}{{e}}}_{{j}}\\ {}& \text{◦}& \text{Equation which must be executed for each basis vector}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{j}}{=}\left[{}\right]{·}{\stackrel{{ˆ}}{{e}}}_{{j}}\\ {}& \text{◦}& {\text{Compute}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{for j =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{1}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{1}}{=}\left[\begin{array}{c}{-}\frac{{3}}{{5}}\\ \frac{{3}}{{5}}\end{array}\right]\\ {}& \text{◦}& \text{1st column of the fundamental matrix}\\ {}& {}& \left[{}\right]{+}\left[{}\right]{=}\left[\begin{array}{c}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ {-}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ {}& \text{◦}& {\text{Compute}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\stackrel{{\to }}{{C}}}_{{j}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{for j =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{2}\\ {}& {}& {\stackrel{{\to }}{{C}}}_{{2}}{=}\left[\begin{array}{c}\frac{{2}}{{5}}\\ \frac{{3}}{{5}}\end{array}\right]\\ {}& \text{◦}& \text{2nd column of the fundamental matrix}\\ {}& {}& \left[{}\right]{+}\left[{}\right]{=}\left[\begin{array}{c}{-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ \frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ {}& {}& \text{Fundamental matrix}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right){=}\left[\begin{array}{cc}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}& {-}\frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{2}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\\ {-}\frac{{3}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}& \frac{{2}{}{{ⅇ}}^{{-}{x}}}{{5}}{+}\frac{{3}{}{{ⅇ}}^{{4}{}{x}}}{{5}}\end{array}\right]\\ \text{▫}& {}& \text{Find a particular solution to the system of ODEs using variation of paramaters}\\ {}& \text{◦}& {\text{Let the particular solution be the fundamental matrix multiplied by}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{and solve for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& \text{◦}& \text{Take the derivative of the particular solution}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\stackrel{{\to }}{{y}}}_{{x}}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Phi }}{}\left({x}\right)\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right)\\ {}& \text{◦}& \text{Substitute particular solution and it's derivative into the system of ODEs}\\ {}& {}& \left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\Phi }}{}\left({x}\right)\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}{A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{The fundamental matrix has columns that are solutions to the homogeneous system so it's derivative follows that of the homogeneous system}\\ {}& {}& {A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}{\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}{A}{·}{\mathrm{\Phi }}{}\left({x}\right){·}\stackrel{{\to }}{{v}}{}\left({x}\right){+}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{Cancel like terms}\\ {}& {}& {\mathrm{\Phi }}{}\left({x}\right){·}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\right){=}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& \text{Multiply by the inverse of the fundamental matrix}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right){=}\left[{}\right]{·}\stackrel{{\to }}{{f}}{}\left({x}\right)\\ {}& \text{◦}& {\text{Integrate to solve for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\stackrel{{\to }}{{v}}{}\left({x}\right)\\ {}& {}& \stackrel{{\to }}{{v}}{}\left({x}\right){=}{{\int }}_{{0}}^{{x}}\left[{}\right]{·}\stackrel{{\to }}{{f}}{}\left({s}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{s}\\ {}& \end{array}$