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•Characteristic polynomial of ODEr3+3r2+4r+2=0•Roots of the characteristic polynomialr=−1,−1−I,−1+I•Solution fromr=−1y1x=ⅇ−x•Solutions fromr=−1−Iandr=−1+Iy2x=ⅇ−xsinx,y3x=ⅇ−xcosx•General solution of the ODEyx=c__1y1x+c__2y2x+c__3y3x•Substitute in solutions and simplifyyx=ⅇ−xc__1+c__2sinx+c__3cosx•Use the initial conditiony0=11=c__1+c__3•Calculate the 1st derivative of the solutionⅆⅆxyx=−ⅇ−xc__1+c__2sinx+c__3cosx+ⅇ−xc__2cosx−c__3sinx•Use the initial conditionⅆⅆxyxx=0|ⅆⅆxyxx=0=−1−1=−c__1−c__3+c__2•Calculate the 2nd derivative of the solutionⅆ2ⅆx2yx=ⅇ−xc__1+c__2sinx+c__3cosx−2ⅇ−xc__2cosx−c__3sinx+ⅇ−x−c__2sinx−c__3cosx•Use the initial conditionⅆ2ⅆx2yxx=0|ⅆ2ⅆx2yxx=0=22=c__1−2c__2•Solve for the unknown coefficientsc__1=2,c__2=0,c__3=−1•Solution to the IVPyx=ⅇ−x2−cosx
macroY=y1x,y2x:
ivpsys2≔diffY,x=`%.`Matrix7,1,`-`4,3,Y,evalY,x=0=1,1
ivpsys2≔ⅆⅆxy1xⅆⅆxy2x=71−43·y1xy2x,y10y20=11
ODEStepsivpsys2
ivpsys3≔diffY,x=Matrix1,2,3,2·Y+1,expx,evalY,x=1=0,−1
ivpsys3≔ⅆⅆxy1xⅆⅆxy2x=y1x+2y2x+13y1x+2y2x+ⅇx,y11y21=0−1
ODEStepsivpsys3
ivpsys4≔diffwx,x=wx+2zx,diffzx,x=3wx+2zx+expx,w−1=2,z−1=−2
ivpsys4≔ⅆⅆxwx=wx+2zx,ⅆⅆxzx=3wx+2zx+ⅇx,w−1=2,z−1=−2
ODEStepsivpsys4
See Also
diff
Int
Student
Student[ODEs]
Student[ODEs][ODESteps]
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