ODE Steps for Systems of ODEs with IVP
Overview
Examples
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.
withStudent:-ODEs:
high_order_ivp1≔diffyx,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_ivp1≔ⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0,ⅆ2ⅆx2yxx=0|ⅆ2ⅆx2yxx=0=2,ⅆⅆxyxx=0|ⅆⅆxyxx=0=−1,y0=1
ODEStepshigh_order_ivp1
Let's solveⅆ3ⅆx3yx+3ⅆ2ⅆx2yx+4ⅆⅆxyx+2yx=0,ⅆ2ⅆx2yxx=0|ⅆ2ⅆx2yxx=0=2,ⅆⅆxyxx=0|ⅆⅆxyxx=0=−1,y0=1•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+I,−1+I−12−I21,−1−I,−1−I−12+I21,−1,−1−11•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→1x=,y→2x=•Consider eigenpair−1,−1−11•Solution to homogeneous system from eigenpairy→3
