 System - Maple Help

Student[ODEs][Solve]

 System
 Solve a system of first order linear ODEs Calling Sequence System(SYS, Y) System(SYS) System(A, F, Y) Parameters

 SYS - list, set, or equation; a system of first order linear ordinary differential equations Y - list or set or Vector of functions; the solving variables A - Matrix; the Matrix of coefficients F - Vector; the Vector of forcing functions Description

 • The System(SYS, Y) command finds the solution of a system of first order linear ODEs.
 • The system SYS may be written as a list or set of ODEs. If the solving variables cannot be unambiguously determined from the form of SYS, Y must also be specified as a list or set containing the solving variables.
 • Alternatively, SYS may be written as a single equation of the form:

$\mathrm{DY}=A·Y+F$

 where Y is a Vector of solving variables, DY a Vector of their derivatives, A is the Matrix of coefficients, and F is the Vector of forcing functions. In this case, Y does not need to be specified as an extra argument since it can be determined from the form of SYS.
 • A third syntax, System(A, F, Y), is also available as a shortcut to the above syntax System(DY = A . Y + F).
 • Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one. Examples

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$

Here the system is written as a set of equations:

 > $\mathrm{sys1}≔\left\{\frac{ⅆ}{ⅆx}{y}_{1}\left(x\right)=7{y}_{1}\left(x\right)+{y}_{2}\left(x\right),\frac{ⅆ}{ⅆx}{y}_{2}\left(x\right)=-4{y}_{1}\left(x\right)+3{y}_{2}\left(x\right)\right\}$
 ${\mathrm{sys1}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{7}{}{{y}}_{{1}}{}\left({x}\right){+}{{y}}_{{2}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{-}{4}{}{{y}}_{{1}}{}\left({x}\right){+}{3}{}{{y}}_{{2}}{}\left({x}\right)\right\}$ (1)
 > $\mathrm{System}\left(\mathrm{sys1},\left\{{y}_{1}\left(x\right),{y}_{2}\left(x\right)\right\}\right)$
 $\left\{{{y}}_{{1}}{}\left({x}\right){=}{-}\frac{{{ⅇ}}^{{5}{}{x}}{}\left({2}{}{\mathrm{_C2}}{}{x}{+}{2}{}{\mathrm{_C1}}{+}{\mathrm{_C2}}\right)}{{4}}{,}{{y}}_{{2}}{}\left({x}\right){=}{{ⅇ}}^{{5}{}{x}}{}\left({\mathrm{_C2}}{}{x}{+}{\mathrm{_C1}}\right)\right\}$ (2)
 > $\mathrm{sys2}≔\left[\frac{ⅆ}{ⅆx}{y}_{1}\left(x\right)=7{y}_{1}\left(x\right)+{y}_{2}\left(x\right)+1,\frac{ⅆ}{ⅆx}{y}_{2}\left(x\right)=-4{y}_{1}\left(x\right)+3{y}_{2}\left(x\right)+{ⅇ}^{x}\right]$
 ${\mathrm{sys2}}{≔}\left[\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{7}{}{{y}}_{{1}}{}\left({x}\right){+}{{y}}_{{2}}{}\left({x}\right){+}{1}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{-}{4}{}{{y}}_{{1}}{}\left({x}\right){+}{3}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\right]$ (3)
 > $\mathrm{System}\left(\mathrm{sys2}\right)$
 $\left\{{{y}}_{{1}}{}\left({x}\right){=}\frac{\left(\left({-}{200}{}{x}{-}{100}\right){}{\mathrm{_C2}}{+}{260}{}{x}{-}{200}{}{\mathrm{_C1}}{+}{23}\right){}{{ⅇ}}^{{5}{}{x}}}{{400}}{+}\frac{{{ⅇ}}^{{x}}}{{16}}{-}\frac{{3}}{{25}}{,}{{y}}_{{2}}{}\left({x}\right){=}\frac{\left(\left({200}{}{\mathrm{_C2}}{-}{260}\right){}{x}{+}{200}{}{\mathrm{_C1}}{+}{107}\right){}{{ⅇ}}^{{5}{}{x}}}{{200}}{-}\frac{{3}{}{{ⅇ}}^{{x}}}{{8}}{-}\frac{{4}}{{25}}\right\}$ (4)
 > $\mathrm{sys3}≔\left\{\frac{ⅆ}{ⅆx}{y}_{1}\left(x\right)=6{y}_{1}\left(x\right)-3{y}_{2}\left(x\right)+1,\frac{ⅆ}{ⅆx}{y}_{2}\left(x\right)=-4{y}_{1}\left(x\right)+9{y}_{2}\left(x\right)+\mathrm{cos}\left(x\right)\right\}$
 ${\mathrm{sys3}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{6}{}{{y}}_{{1}}{}\left({x}\right){-}{3}{}{{y}}_{{2}}{}\left({x}\right){+}{1}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{-}{4}{}{{y}}_{{1}}{}\left({x}\right){+}{9}{}{{y}}_{{2}}{}\left({x}\right){+}{\mathrm{cos}}{}\left({x}\right)\right\}$ (5)
 > $\mathrm{System}\left(\mathrm{sys3}\right)$
 $\left\{{{y}}_{{1}}{}\left({x}\right){=}\frac{\left(\left({380247}{}{\mathrm{_C1}}{+}{59492}\right){}\sqrt{{57}}{+}{1140741}{}{\mathrm{_C1}}{+}{424080}\right){}{{ⅇ}}^{{-}\frac{\left({-}{15}{+}\sqrt{{57}}\right){}{x}}{{2}}}}{{3041976}}{+}\frac{\left(\left({-}{380247}{}{\mathrm{_C2}}{-}{59492}\right){}\sqrt{{57}}{+}{1140741}{}{\mathrm{_C2}}{+}{424080}\right){}{{ⅇ}}^{\frac{\left({15}{+}\sqrt{{57}}\right){}{x}}{{2}}}}{{3041976}}{-}\frac{{123}{}{\mathrm{cos}}{}\left({x}\right)}{{1906}}{+}\frac{{45}{}{\mathrm{sin}}{}\left({x}\right)}{{1906}}{-}\frac{{3}}{{14}}{,}{{y}}_{{2}}{}\left({x}\right){=}\frac{\left({1520988}{}{\mathrm{_C1}}{+}{20467}{}\sqrt{{57}}{+}{176567}\right){}{{ⅇ}}^{{-}\frac{\left({-}{15}{+}\sqrt{{57}}\right){}{x}}{{2}}}}{{1520988}}{+}\frac{\left({1520988}{}{\mathrm{_C2}}{-}{20467}{}\sqrt{{57}}{+}{176567}\right){}{{ⅇ}}^{\frac{\left({15}{+}\sqrt{{57}}\right){}{x}}{{2}}}}{{1520988}}{-}\frac{{261}{}{\mathrm{cos}}{}\left({x}\right)}{{1906}}{+}\frac{{49}{}{\mathrm{sin}}{}\left({x}\right)}{{1906}}{-}\frac{{2}}{{21}}\right\}$ (6)

In these examples the systems are written in Vector-Matrix format:

 > $Y≔⟨v\left(x\right),w\left(x\right)⟩$
 ${Y}{≔}\left[\begin{array}{c}{v}{}\left({x}\right)\\ {w}{}\left({x}\right)\end{array}\right]$ (7)
 > $A≔⟨⟨7|1⟩,⟨-4|3⟩⟩$
 ${A}{≔}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]$ (8)
 > $F≔⟨1,{ⅇ}^{x}⟩$
 ${F}{≔}\left[\begin{array}{c}{1}\\ {{ⅇ}}^{{x}}\end{array}\right]$ (9)
 > $\mathrm{sys4}≔\frac{\partial }{\partial x}Y=\mathrm{.}\left(A,Y\right)$
 ${\mathrm{sys4}}{≔}\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}{7}{}{v}{}\left({x}\right){+}{w}{}\left({x}\right)\\ {-}{4}{}{v}{}\left({x}\right){+}{3}{}{w}{}\left({x}\right)\end{array}\right]$ (10)
 > $\mathrm{System}\left(\mathrm{sys4}\right)$
 $\left\{{v}{}\left({x}\right){=}{-}\frac{{{ⅇ}}^{{5}{}{x}}{}\left({2}{}{\mathrm{_C2}}{}{x}{+}{2}{}{\mathrm{_C1}}{+}{\mathrm{_C2}}\right)}{{4}}{,}{w}{}\left({x}\right){=}{{ⅇ}}^{{5}{}{x}}{}\left({\mathrm{_C2}}{}{x}{+}{\mathrm{_C1}}\right)\right\}$ (11)
 > $\mathrm{sys5}≔\frac{\partial }{\partial x}Y=\mathrm{%.}\left(A,Y\right)+F$
 ${\mathrm{sys5}}{≔}\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({x}\right)\end{array}\right]{=}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%.}}{}\left({\mathrm{RTABLE}}{}\left({36893628488179307692}{,}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]{,}{\mathrm{Matrix}}\right){,}\left[\begin{array}{c}v{}\left(x\right)\\ w{}\left(x\right)\end{array}\right]\right)\right]\right){+}\left[\begin{array}{c}{1}\\ {{ⅇ}}^{{x}}\end{array}\right]$ (12)
 > $\mathrm{System}\left(\mathrm{sys5}\right)$
 $\left\{{v}{}\left({x}\right){=}\frac{\left(\left({-}{200}{}{x}{-}{100}\right){}{\mathrm{_C2}}{+}{260}{}{x}{-}{200}{}{\mathrm{_C1}}{+}{23}\right){}{{ⅇ}}^{{5}{}{x}}}{{400}}{+}\frac{{{ⅇ}}^{{x}}}{{16}}{-}\frac{{3}}{{25}}{,}{w}{}\left({x}\right){=}\frac{\left(\left({200}{}{\mathrm{_C2}}{-}{260}\right){}{x}{+}{200}{}{\mathrm{_C1}}{+}{107}\right){}{{ⅇ}}^{{5}{}{x}}}{{200}}{-}\frac{{3}{}{{ⅇ}}^{{x}}}{{8}}{-}\frac{{4}}{{25}}\right\}$ (13)
 > $B≔⟨⟨1|2⟩,⟨3|2⟩⟩$
 ${B}{≔}\left[\begin{array}{cc}{1}& {2}\\ {3}& {2}\end{array}\right]$ (14)
 > $\mathrm{sys6}≔\frac{\partial }{\partial x}Y=\mathrm{.}\left(B,Y\right)+F$
 ${\mathrm{sys6}}{≔}\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}{v}{}\left({x}\right){+}{2}{}{w}{}\left({x}\right){+}{1}\\ {3}{}{v}{}\left({x}\right){+}{2}{}{w}{}\left({x}\right){+}{{ⅇ}}^{{x}}\end{array}\right]$ (15)
 > $\mathrm{System}\left(B,F,Y\right)$
 $\left\{{v}{}\left({x}\right){=}\frac{\left({-}{30}{}{\mathrm{_C1}}{-}{12}\right){}{{ⅇ}}^{{-}{x}}}{{30}}{+}\frac{\left({20}{}{\mathrm{_C2}}{+}{7}\right){}{{ⅇ}}^{{4}{}{x}}}{{30}}{-}\frac{{{ⅇ}}^{{x}}}{{3}}{+}\frac{{1}}{{2}}{,}{w}{}\left({x}\right){=}\frac{\left({20}{}{\mathrm{_C1}}{+}{8}\right){}{{ⅇ}}^{{-}{x}}}{{20}}{-}\frac{{3}}{{4}}{+}\frac{\left({20}{}{\mathrm{_C2}}{+}{7}\right){}{{ⅇ}}^{{4}{}{x}}}{{20}}\right\}$ (16) Compatibility

 • The Student[ODEs][Solve][System] command was introduced in Maple 2021.