SystemsOfODEsWithIVP - Maple Help

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ODE Steps for Systems of ODEs with IVP

 

Overview

Examples

Overview

• 

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

• 

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 solve&DifferentialD;3&DifferentialD;x3yx+3&DifferentialD;2&DifferentialD;x2yx+4&DifferentialD;&DifferentialD;xyx+2yx=0&comma;&DifferentialD;2&DifferentialD;x2yxx=0|&DifferentialD;2&DifferentialD;x2yxx=0=2&comma;&DifferentialD;&DifferentialD;xyxx=0|&DifferentialD;&DifferentialD;xyxx=0=−1&comma;y0=1Highest derivative means the order of the ODE is3&DifferentialD;3&DifferentialD;x3yxConvert linear ODE into a system of first order ODEsDefine new variabley1xy1x=yxDefine new variabley2xy2x=&DifferentialD;&DifferentialD;xyxDefine new variabley3xy3x=&DifferentialD;2&DifferentialD;x2yxIsolate for&DifferentialD;&DifferentialD;xy3xusing original ODE&DifferentialD;&DifferentialD;xy3x=3y3x4y2x2y1xConvert linear ODE into a system of first order ODEsy2x=&DifferentialD;&DifferentialD;xy1x&comma;y3x=&DifferentialD;&DifferentialD;xy2x&comma;&DifferentialD;&DifferentialD;xy3x=3y3x4y2x2y1xDefine vectoryx=y3xy1xy2xSystem to solve&DifferentialD;&DifferentialD;xyx=A·yxTo solve the system find eigenvalues and eigenvectors ofAA=−3−2−4001100Eigenpairs of A−1+I&comma;−1+I12I21&comma;−1I&comma;−1I12+I21&comma;−1&comma;−1−11Consider complex eigenpair, complex conjugate eigenvalue can be ignored−1+I&comma;−1+I12I21Solution from eigenpairUse Euler identity to write solution in terms of sin and cosSimplify expression<