ODE Steps for Systems of ODEs
Overview
Examples
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.
withStudent:-ODEs:
high_order_ode1≔diffyx,x,x,x+3diffyx,x,x+4diffyx,x+2yx=0
high_order_ode1≔ⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0
ODEStepshigh_order_ode1
Let's solveⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0•Highest derivative means the order of the ODE is3ⅆ3ⅆx3yx▫Convert linear ODE into a system of first order ODEs◦Define new variabley1xy1x=yx◦Define new variabley2xy2x=ⅆⅆxyx◦Define new variabley3xy3x=ⅆ2ⅆx2yx◦Isolate forⅆⅆxy3xusing original ODEⅆⅆxy3x=−3y3x−4y2x−2y1xConvert linear ODE into a system of first order ODEsy2x=ⅆⅆxy1x,y3x=ⅆⅆxy2x,ⅆⅆxy3x=−3y3x−4y2x−2y1x•Define vectory→x=y3xy1xy2x•System to solveⅆⅆxy→x=A·y→x•To solve the system find eigenvalues and eigenvectors ofAA=−3−2−4001100•Eigenpairs of A−1,−1−11,−1+I,−1+I−12−I21,−1−I,−1−I−12+I21•Consider eigenpair−1,−1−11•Solution to homogeneous system from eigenpairy→1x=•Consider complex eigenpair, complex conjugate eigenvalue can be ignored−1+I,−1+I−12−I21•Solution from eigenpair•Use Euler identity to write solution in terms of sin and cos•Simplify expression•Both real and imaginary parts are solutions to the homogeneous systemy→2x=,y→3x=•General solution to the system of ODEsy→x=_C1y→1x+_C2y→2x+_C3y→3x•Substitute solutions into the general solutiony→x=++•First component of the vector is the solution to the ODEyx=−ⅇ−x_C2+_C3sinx+_C2−_C3cosx+_C1
macroY=y→:
sys2≔diffYx,x=Matrix7,1,−4,3·Yx
sys2≔ⅆⅆxy→x=71−43·y→x
ODEStepssys2
Let's solveⅆⅆxy→x=71−43·y→x•System to solveⅆⅆxy→x=A·y→x•To solve the system find eigenvalues and eigenvectors ofAA=71−43•Eigenpairs of A5,−121,5,00•Consider eigenpair, with eigenvalue of algebraic multiplicity 25,−121•First solution from eigenvalue5y→1x=•Form of the 2nd homogeneous solution wherep→is to be solved for,λ=5is the eigenvalue, andv→is the eigenvectory→2x=ⅇλxxv→+p→+v→•Note that thexmultiplyingv→makes this solution linearly independent to the 1st solution obtained fromλ=5•substitutey→2xinto the systemλⅇλxxv→+p→+v→+ⅇλxv→=ⅇλxA·xv→+p→+v→•Use the fact thatv→is an eigenvector ofAλⅇλxxv→+p→+v→+ⅇλxv→=ⅇλxλv→+λxv→+A·p→•Simplify equationλp→+v→=A·p→•Make use of the identity matrixIλI·p→+v→=A·p→•Conditionp→must meet fory→2xto be a solution to the system−λI+A·p→=v→•Choosep→to use in the second solution to the system from eigenvalue5·p→=−121•Choice ofp→p→=−140•second solution from eigenvalue5y→2x=•General solution to the system of ODEsy→x=_C1y→1x+_C2y→2x•Substitute solutions into the general solutiony→x=+
sys3≔diffy1x,x,diffy2x,x=Matrix1,2,3,2·y1x,y2x+1,expx
sys3≔ⅆⅆxy1xⅆⅆxy2x=y1x+2y2x+13y1x+2y2x+ⅇx
ODEStepssys3
Let's solveⅆⅆxy1xⅆⅆxy2x=y1x+2y2x+13y1x+2y2x+ⅇx•Define vectory→x=y1xy2x•System to solveⅆⅆxy→x=1232·y→x+1ⅇx•Define forcing functionf→x=1ⅇx•To solve the system find eigenvalues and eigenvectors ofAA=1232•Eigenpairs of A4,231,−1,−11•Consider eigenpair4,231•Solution to homogeneous system from eigenpairy→1x=•Consider eigenpair−1,−11•Solution to homogeneous system from eigenpairy→2x=•General solution to the system of ODEs wherey→pxis a particular solution to the systemy→x=_C1y→1x+_C2y→2x+y→px▫Fundamental matrix◦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 atx=0Φx◦Compute the solution to the homogeneous system for each basis vector of the standard basis being the inital condition atx=0ⅆⅆxy→x=A·y→x◦LetBbe the matrix with solutions to the homogeneous system as columns evaluated atx=0B=23−111◦For each basis vector we need to solve the system of linear equations forC→jwhich contains the values to multiply each homogeneous solution by to get the basis vector as the intital conditionB·C→j=eˆj◦Equation which must be executed for each basis vectorC→j=·eˆj◦ComputeC→jfor j =1C→1=35−35◦1st column of the fundamental matrix+=3ⅇ−x5+2ⅇ4x5−3ⅇ−x5+3ⅇ4x5◦ComputeC→jfor j =2C→2=3525◦2nd column of the fundamental matrix+=−2ⅇ−x5+2ⅇ4x52ⅇ−x5+3ⅇ4x5Fundamental matrixΦx=3ⅇ−x5+2ⅇ4x5−2ⅇ−x5+2ⅇ4x5−3ⅇ−x5+3ⅇ4x52ⅇ−x5+3ⅇ4x5▫Find a particular solution to the system of ODEs using variation of paramaters◦Let the particular solution be the fundamental matrix multiplied byv→xand solve forv→xy→xx=Φx·v→x◦Take the derivative of the particular solutionⅆⅆxy→xx=ⅆⅆxΦx·v→x+Φx·ⅆⅆxv→x◦Substitute particular solution and it's derivative into the system of ODEsⅆⅆxΦx·v→x+Φx·ⅆⅆxv→x=A·Φx·v→x+f→x◦The fundamental matrix has columns that are solutions to the homogeneous system so it's derivative follows that of the homogeneous systemA·Φx·v→x+Φx·ⅆⅆxv→x=A·Φx·v→x+f→x◦Cancel like termsΦx·ⅆⅆxv→x=f→x◦Multiply by the inverse of the fundamental matrixⅆⅆxv→x=·f→x◦Integrate to solve forv→xv→x=∫0x·f→sⅆs◦Plugv→xinto the equation for the particular solutiony→xx=Φx·∫0x·f→sⅆs◦Plug in the fundamental matrix and the forcing function and computey→xx=7ⅇ4x30−ⅇx3+12−2ⅇ−x57ⅇ4x20−34+2ⅇ−x5Find a particular solution to the system of ODEs using variation of paramatersy→xx=7ⅇ4x30−ⅇx3+12−2ⅇ−x57ⅇ4x20−34+2ⅇ−x5•Plug particular solution back into general solutiony→x=_C1y→1x+_C2y→2x+7ⅇ4x30−ⅇx3+12−2ⅇ−x57ⅇ4x20−34+2ⅇ−x5•Solution to the system of ODEsy1xy2x=−30_C2−12ⅇ−x30+20_C1+7ⅇ4x30−ⅇx3+1220_C2+8ⅇ−x20−34+20_C1+7ⅇ4x20
sys4≔diffy1x,x=y1x+2y2x,diffy2x,x=3y1x+2y2x+expx
sys4≔ⅆⅆxy1x=y1x+2y2x,ⅆⅆxy2x=3y1x+2y2x+ⅇx
ODEStepssys4
Let's solveⅆⅆxy1x=y1x+2y2x,ⅆⅆxy2x=3y1x+2y2x+ⅇx•Define vectory→x=y1xy2x•Convert system into a vector equationⅆⅆxy→x=+0ⅇx•System to solveⅆⅆxy→x=1232·y→x+0ⅇx•Define forcing functionf→x=0ⅇx•To solve the system find eigenvalues and eigenvectors ofAA=1232•Eigenpairs of A−1,−11,4,231•Consider eigenpair−1,−11•Solution to homogeneous system from eigenpairy→1x=•Consider eigenpair4,231•Solution to homogeneous system from eigenpairy→2x=•General solution to the system of ODEs wherey→pxis a particular solution to the systemy→x=_C1y→1x+_C2y→2x+y→px▫Fundamental matrix◦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 atx=0Φx◦Compute the solution to the homogeneous system for each basis vector of the standard basis being the inital condition atx=0ⅆⅆxy→x=A·y→x◦LetBbe the matrix with solutions to the homogeneous system as columns evaluated atx=0B=−12311◦For each basis vector we need to solve the system of linear equations forC→jwhich contains the values to multiply each homogeneous solution by to get the basis vector as the intital conditionB·C→j=eˆj◦Equation which must be executed for each basis vectorC→j=·eˆj◦ComputeC→jfor j =1C→1=−3535◦1st column of the fundamental matrix+=3ⅇ−x5+2ⅇ4x5−3ⅇ−x5+3ⅇ4x5◦ComputeC→jfor j =2C→2=2535◦2nd column of the fundamental matrix+=−2ⅇ−x5+2ⅇ4x52ⅇ−x5+3ⅇ4x5Fundamental matrixΦx=3ⅇ−x5+2ⅇ4x5−2ⅇ−x5+2ⅇ4x5−3ⅇ−x5+3ⅇ4x52ⅇ−x5+3ⅇ4x5▫Find a particular solution to the system of ODEs using variation of paramaters◦Let the particular solution be the fundamental matrix multiplied byv→xand solve forv→xy→xx=Φx·v→x◦Take the derivative of the particular solutionⅆⅆxy→xx=ⅆⅆxΦx·v→x+Φx·ⅆⅆxv→x◦Substitute particular solution and it's derivative into the system of ODEsⅆⅆxΦx·v→x+Φx·ⅆⅆxv→x=A·Φx·v→x+f→x◦The fundamental matrix has columns that are solutions to the homogeneous system so it's derivative follows that of the homogeneous systemA·Φx·v→x+Φx·ⅆⅆxv→x=A·Φx·v→x+f→x◦Cancel like termsΦx·ⅆⅆxv→x=f→x◦Multiply by the inverse of the fundamental matrixⅆⅆxv→x=·f→x◦Integrate to solve forv→xv→x=∫0x·f→sⅆs
