SystemsOfODEsWithIVP - Maple Help

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ODE Steps for Systems of ODEs with IVP

 

Overview

Examples

Overview

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This help page gives a few examples of using the command ODESteps to solve systems of ordinary differential equations with initial values.

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See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

withStudent:-ODEs:

high_order_ivp1diffyx,x,x,x+3diffyx,x,x+4diffyx,x+2yx=0,evaldiffyx,x,x=0=1,evaldiffyx,x,x,x=0=2,y0=1

high_order_ivp1ⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0,ⅆ2ⅆx2yxx=0|ⅆ2ⅆx2yxx=0=2,ⅆⅆxyxx=0|ⅆⅆxyxx=0=−1,y0=1

(1)

ODEStepshigh_order_ivp1

Let's solveⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0,ⅆ2ⅆx2yxx=0|ⅆ2ⅆx2yxx=0=2,ⅆⅆxyxx=0|ⅆⅆxyxx=0=−1,y0=1Highest derivative means the order of the ODE is3ⅆ3ⅆx3yxConvert linear ODE into a system of first order ODEsDefine new variabley1xy1x=yxDefine new variabley2xy2x=ⅆⅆxyxDefine new variabley3xy3x=ⅆ2ⅆx2yxIsolate forⅆⅆxy3xusing original ODEⅆⅆxy3x=3y3x4y2x2y1xConvert linear ODE into a system of first order ODEsy2x=ⅆⅆxy1x,y3x=ⅆⅆxy2x,ⅆⅆxy3x=3y3x4y2x2y1xDefine vectoryx=y3xy1xy2xSystem to solveⅆⅆxyx=A·yxTo solve the system find eigenvalues and eigenvectors ofAA=−3−2−4001100Eigenpairs of A−1+I,−1+I12I21,−1I,−1I12+I21,−1,−1−11Consider complex eigenpair, complex conjugate eigenvalue can be ignored−1+I,−1+I12I21Solution from eigenpairUse Euler identity to write solution in terms of sin and cos