Second Order ODEs - Maple Help
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ODE Steps for Second Order ODEs

 

Overview

Examples

Overview

• 

This help page gives a few examples of using the command ODESteps to solve second order ordinary differential equations.

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

Examples

withStudent:-ODEs:

ode12xdiffyx,x9x2+2diffyx,x+x2+1diffyx,x,x=0

ode12xⅆⅆxyx9x2+2ⅆⅆxyx+x2+1ⅆ2ⅆx2yx=0

(1)

ODEStepsode1

Let's solve2xⅆⅆxyx9x2+2ⅆⅆxyx+x2+1ⅆ2ⅆx2yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxMake substitutionu=ⅆⅆxyxto reduce order of ODE2xux9x2+2ux+x2+1ⅆⅆxux=0Check if ODE is exactODE is exact if the lhs is the total derivative of aC2functionⅆⅆxFx,ux=0Compute derivative of lhsxFx,u+uFx,uⅆⅆxux=0Evaluate derivatives2x=2xCondition met, ODE is exactExact ODE implies solution will be of this formFx,u=C1,Mx,u=xFx,u,Nx,u=uFx,uSolve forFx,uby integratingMx,uwith respect toxFx,u=2ux9x2ⅆx+_F1uEvaluate integralFx,u=x2u3x3+_F1uTake derivative ofFx,uwith respect touNx,u=uFx,uCompute derivativex2+2u+1=x2+ⅆⅆu_F1uIsolate forⅆⅆu_F1uⅆⅆu_F1u=2u+1Solve for_F1u_F1u=u2+uSubstitute_F1uinto equation forFx,uFx,u=x2u3x3+u2+uSubstituteFx,uinto the solution of the ODEx2u3x3+u2+u=C1Solve foruxux=x2212x4+12x3+2x2+4C1+12,ux=x2212+x4+12x3+2x2+4C1+12Solve 1st ODE foruxux=x2212x4+12x3+2x2+4C1+12Make substitutionu=ⅆⅆxyxⅆⅆxyx=x2212x4+12x3+2x2+4C1+12Integrate both sides to solve foryxⅆⅆxyxⅆx=x2212x4+12x3+2x2+4C1+12ⅆx+C2Compute lhsyx=x2212x4+12x3+2x2+4C1+12ⅆx+C2Solve 2nd ODE foruxux=x2212+x4+12x3+2x2+4C1+12Make substitutionu=ⅆⅆxyxⅆⅆxyx=x2212+x4+12x3+2x2+4C1+12Integrate both sides to solve foryxⅆⅆxyxⅆx=x2212+x4+12x3+2x2+4C1+12ⅆx+C2Compute lhsyx=x2212+x4+12x3+2x2+4C1+12ⅆx+C2

(2)

ode2diffyx,x,xdiffyx,xxexpx=0

ode2ⅆ2ⅆx2yxⅆⅆxyxxⅇx=0

(3)

ODEStepsode2

Let's solveⅆ2ⅆx2yxⅆⅆxyxxⅇx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=ⅆⅆxyx+xⅇxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yxⅆⅆxyx=xⅇxCharacteristic polynomial of homogeneous ODEr2r=0Factor the characteristic polynomialrr1=0Roots of the characteristic polynomialr=0,11st solution of the homogeneous ODEy1x=12nd solution of the homogeneous ODEy2x=ⅇxGeneral solution of the ODEyx=C1y1x+C2y2x+ypxSubstitute in solutions of the homogeneous ODEyx=C1+C2ⅇx+ypxFind a particular solutionypxof the ODEUse variation of parameters to findypherefxis the forcing functionypx=y1xy2xfxWy1x,y2xⅆx+y2xy1xfxWy1x,y2xⅆx,fx=xⅇxWronskian of solutions of the homogeneous equationWy1x,y2x=1ⅇx0ⅇxCompute WronskianWy1x,y2x=ⅇxSubstitute functions into equation forypxypx=xⅇxⅆx+ⅇxxⅆxCompute integralsypx=ⅇx1x+12x2Substitute particular solution into general solution to ODEyx=C1+C2ⅇx+ⅇx1x+12x2

(4)

ode3diffyx,x,x+5diffyx,x2yx=0

ode3ⅆ2ⅆx2yx+5ⅆⅆxyx2yx=0

(5)

ODEStepsode3

Let's solveⅆ2ⅆx2yx+5ⅆⅆxyx2yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxDefine new dependent variableuux=ⅆⅆxyxComputeⅆ2ⅆx2yxⅆⅆxux=ⅆ2ⅆx2yxUse chain rule on the lhsⅆⅆxyxⅆⅆyuy=ⅆ2ⅆx2yxSubstitute in the definition ofuuyⅆⅆyuy=ⅆ2ⅆx2yxMake substitutionsⅆⅆxyx=uy,ⅆ2ⅆx2yx=uyⅆⅆyuyto reduce order of ODEuyⅆⅆyuy+5uy2y=0Separate variablesⅆⅆyuyuy=5yIntegrate both sides with respect toyⅆⅆyuyuyⅆy=5yⅆy+C1Evaluate integrallnuy=5lny+C1Solve foruyuy=ⅇC1y5Solve 1st ODE foruyuy=ⅇC1y5Revert to original variables with substitutionuy=ⅆⅆxyx,y=yxⅆⅆxyx=ⅇC1yx5Separate variablesⅆⅆxyxyx5=ⅇC1Integrate both sides with respect toxⅆⅆxyxyx5ⅆx=ⅇC1ⅆx+C2Evaluate integralyx66=ⅇC1x+C2Solve foryxyx=6ⅇC1x+6C216,yx=6ⅇC1x+6C216

(6)

ode4diffyx,x,xdiffyx,x6yx=0

ode4ⅆ2ⅆx2yxⅆⅆxyx6yx=0

(7)

ODEStepsode4

Let's solveⅆ2ⅆx2yxⅆⅆxyx6yx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxCharacteristic polynomial of ODEr2r6=0Factor the characteristic polynomialr+2r3=0Roots of the characteristic polynomialr=−2,31st solution of the ODEy1x=ⅇ2x2nd solution of the ODEy2x=ⅇ3xGeneral solution of the ODEyx=C1y1x+C2y2xSubstitute in solutionsyx=C1ⅇ2x+C2ⅇ3x

(8)

ode5diffyx,x,xdiffyx,x=x2+6yx

ode5ⅆ2ⅆx2yxⅆⅆxyx=x2+6yx

(9)

ODEStepsode5

Let's solveⅆ2ⅆx2yxⅆⅆxyx=x2+6yxHighest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=ⅆⅆxyx+x2+6yxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yxⅆⅆxyx6yx=x2Characteristic polynomial of homogeneous ODEr2r6=0Factor the characteristic polynomialr+2r3=0Roots of the characteristic polynomialr=−2,31st solution of the homogeneous ODEy1x=ⅇ2x2nd solution of the homogeneous ODEy2x=ⅇ3xGeneral solution of the ODEyx=C1y1x+C2y2x+ypxSubstitute in solutions of the homogeneous ODEyx=C1ⅇ2x+C2ⅇ3x+ypxFind a particular solutionypxof the ODEUse variation of parameters to findypherefxis the forcing functionypx=y1xy2xfxWy1x,y2xⅆx+y2xy1xfxWy1x,y2xⅆx,fx=x2Wronskian of solutions of the homogeneous equationWy1x,y2x=ⅇ2xⅇ3x2ⅇ2x3ⅇ3xCompute WronskianWy1x,y2x=5ⅇxSubstitute functions into equation forypxypx=ⅇ5xx2ⅇ3xⅆxx2ⅇ2xⅆxⅇ2x5Compute integralsypx=16x2+118x7108Substitute particular solution into general solution to ODEyx=C1ⅇ2x+C2ⅇ3xx26+x187108

(10)

ode6diffyx,x,x+4yx=4diffyx,x

ode6ⅆ2ⅆx2yx+4yx=4ⅆⅆxyx

(11)

ODEStepsode6

Let's solveⅆ2ⅆx2yx+4yx=4ⅆⅆxyxHighest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=4yx4ⅆⅆxyxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yx+4yx+4ⅆⅆxyx=0Characteristic polynomial of ODEr2+4r+4=0Factor the characteristic polynomialr+22=0Root of the characteristic polynomialr=−21st solution of the ODEy1x=ⅇ2xRepeated root, multiplyy1xbyxto ensure linear independencey2x=xⅇ2xGeneral solution of the ODEyx=C1y1x+C2y2xSubstitute in solutionsyx=C1ⅇ2x+C2xⅇ2x

(12)

ode75diffyx,x,x+20yx+15sinx=20diffyx,x

ode75ⅆ2ⅆx2yx+20yx+15sinx=20ⅆⅆxyx

(13)

ODEStepsode7

Let's solve5ⅆ2ⅆx2yx+20yx+15sinx=20ⅆⅆxyxHighest derivative means the order of the ODE is2ⅆ2ⅆx2yxIsolate 2nd derivativeⅆ2ⅆx2yx=4yx3sinx4ⅆⅆxyxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODE; ODE is linearⅆ2ⅆx2yx+4yx+4ⅆⅆxyx=3sinxCharacteristic polynomial of homogeneous ODEr2+4r+4=0Factor the characteristic polynomialr+22=0Root of the characteristic polynomialr=−21st solution of the homogeneous ODEy1x=ⅇ2xRepeated root, multiplyy1xbyxto ensure linear independencey2x=xⅇ2xGeneral solution of the ODEyx=C1y1x+C2y2x+ypxSubstitute in solutions of the homogeneous ODEyx=C1ⅇ2x+C2xⅇ2x+ypxFind a particular solutionypxof the ODEUse variation of parameters to findypherefxis the forcing functionypx=y1xy2xfxWy1x,y2xⅆx+y2xy1xfxWy1x,y2xⅆx,fx=3sinxWronskian of solutions of the homogeneous equationWy1x,y2x=ⅇ2xxⅇ2x2ⅇ2xⅇ2x2xⅇ2xCompute WronskianWy1x,y2x=ⅇ4xSubstitute functions into equation forypxypx=3ⅇ2xsinxxⅇ2xⅆxxsinxⅇ2xⅆxCompute integralsypx=12cosx259sinx25Substitute particular solution into general solution to ODEyx=C1ⅇ2x+C2xⅇ2x+12cosx259sinx25

(14)

ode8diffyx,x,x+2yx+2diffyx,x=0

ode8ⅆ2ⅆx2yx+2yx+2ⅆⅆxyx=0

(15)

ODEStepsode8

Let's solveⅆ2ⅆx2yx+2yx+2ⅆⅆxyx=0Highest derivative means the order of the ODE is2ⅆ2ⅆx2yxCharacteristic polynomial of ODEr2+2r+2=0Use quadratic formula to solve forrr=−2±−42Roots of the characteristic polynomialr=−1I,−1+I1st solution of the ODEy1x=ⅇxcosx2nd solution of the ODEy2x=ⅇxsinxGeneral solution of the ODEyx=C1y1x+C2y2xSubstitute in solutionsyx=C1ⅇxcosx+C2ⅇxsinx

(16)

ode9diffyx,x,x+2yx2diffyx,x=expx

ode9ⅆ2ⅆx2yx+2yx2ⅆⅆxyx=ⅇx

(17)

ODEStepsode9

Let's solveⅆ2ⅆx2yx+2yx2ⅆⅆxyx=ⅇxHighest derivative means the order of the ODE is2ⅆ2ⅆx2yxCharacteristic polynomial of homogeneous ODEr22r+2=0Use quadratic formula to solve forrr=2±−42Roots of the characteristic polynomialr=1I,1+I1st solution of the homogeneous ODEy1x=ⅇxcosx2nd solution of the homogeneous ODEy2x=ⅇxsinxGeneral solution of the ODEyx=C1y1x+C2y2x+ypxSubstitute in solutions of the homogeneous ODEyx=C1ⅇxcosx+C2ⅇxsinx+ypxFind a particular solutionypxof the ODEUse variation of parameters to findypherefxis the forcing functionypx=y1xy2xfxWy1x,y2xⅆx+y2xy1xfxWy1x,y2xⅆx,fx=ⅇxWronskian of solutions of the homogeneous equationWy1x,y2x=ⅇxcosxⅇxsinxⅇxcosxⅇxsinxⅇxsinx+ⅇxcosxCompute WronskianWy1x,y2x=ⅇ2xSubstitute functions into equation forypxypx=ⅇxcosxsinxⅆx+sinxcosxⅆxCompute integralsypx=ⅇxSubstitute particular solution into general solution to ODEyx=C1ⅇxcosx+C2ⅇxsinx+ⅇx

(18)

See Also

diff

Int

Student

Student[ODEs]

Student[ODEs][ODESteps]

 


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