First Order IVPs - Maple Help
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ODE Steps for First Order IVPs

 

Overview

Examples

Overview

• 

This help page gives a few examples of using the command ODESteps to solve first order initial value problems.

• 

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

Examples

withStudent:-ODEs:

ivp1t2zt+1+zt2t1diffzt,t=0,z3=1

ivp1t2zt+1+zt2t1ⅆⅆtzt=0,z3=1

(1)

ODEStepsivp1

Let's solvet2zt+1+zt2t1ⅆⅆtzt=0,z3=1Highest derivative means the order of the ODE is1ⅆⅆtztSolve for the highest derivativeⅆⅆtzt=t2zt+1zt2t1Separate variablesⅆⅆtztzt2zt+1=t2t1Integrate both sides with respect totⅆⅆtztzt2zt+1ⅆt=t2t1ⅆt+c__1Evaluate integralzt22zt+lnzt+1=t22tlnt1+c__1Use initial conditionz3=112+ln2=152ln2+c__1Solve forc__1c__1=7+2ln2Substitutec__1=7+2ln2into general solution and simplifyzt22zt+lnzt+1=t22tlnt1+7+2ln2Solution to the IVPzt22zt+lnzt+1=t22tlnt1+7+2ln2

(2)

ivp22xyx9x2+2yx+x2+1diffyx,x=0,y0=1

ivp22xyx9x2+2yx+x2+1ⅆⅆxyx=0,y0=1

(3)

ODEStepsivp2

Let's solve2xyx9x2+2yx+x2+1ⅆⅆxyx=0,y0=1Highest derivative means the order of the ODE is1ⅆⅆxyxCheck if ODE is exactODE is exact if the lhs is the total derivative of aC2functionⅆⅆxGx,yx=0Compute derivative of lhsxGx,y+yGx,yⅆⅆxyx=0Evaluate derivatives2x=2xCondition met, ODE is exactExact ODE implies solution will be of this formGx,y=c__1,Mx,y=xGx,y,Nx,y=yGx,ySolve forGx,yby integratingMx,ywith respect toxGx,y=9x2+2xyⅆx+_F1yEvaluate integralGx,y=3x3+x2y+_F1yTake derivative ofGx,ywith respect toyNx,y=yGx,yCompute derivativex2+2y+1=x2+ⅆⅆy_F1yIsolate forⅆⅆy_F1yⅆⅆy_F1y=2y+1Solve for_F1y_F1y=y2+ySubstitute_F1yinto equation forGx,yGx,y=3x3+x2y+y2+ySubstituteGx,yinto the solution of the ODE3x3+x2y+y2+y=c__1Solve foryxyx=x2212x4+12x3+2x2+4c__1+12,yx=x2212+x4+12x3+2x2+4c__1+12Use initial conditiony0=11=124c__1+12Solve forc__1No solutionSolution does not satisfy initial conditionUse initial conditiony0=11=12+4c__1+12Solve forc__1c__1=2Substitutec__1=2into general solution and simplifyyx=x2212+x4+12x3+2x2+92Solution to the IVPyx=x2212+x4+12x3+2x2+92

(4)

ivp3diffyx,xyxxexpx=0,ya=b

ivp3ⅆⅆxyxyxxⅇx=0,ya=b

(5)

ODEStepsivp3

Let's solveⅆⅆxyxyxxⅇx=0,ya=bHighest derivative means the order of the ODE is1ⅆⅆxyxSolve for the highest derivativeⅆⅆxyx=yx+xⅇxGroup terms withyxon the lhs of the ODE and the rest on the rhs of the ODEⅆⅆxyxyx=xⅇxThe ODE is linear; multiply by an integrating factorμxμxⅆⅆxyxyx=μxxⅇxAssume the lhs of the ODE is the total derivativeⅆⅆxyxμxμxⅆⅆxyxyx=ⅆⅆxyxμx+yxⅆⅆxμxIsolateⅆⅆxμxⅆⅆxμx=μxSolve to find the integrating factorμx=ⅇxIntegrate both sides with respect toxⅆⅆxyxμxⅆx=μxxⅇxⅆx+c__1Evaluate the integral on the lhsyxμx=μxxⅇxⅆx+c__1Solve foryxyx=μxxⅇxⅆx+c__1μxSubstituteμx=ⅇxyx=ⅇxxⅇxⅆx+c__1ⅇxEvaluate the integrals on the rhsyx=x22+c__1ⅇxSimplifyyx=ⅇxx22+c__1Use initial conditionya=bb=ⅇaa22+c__1Solve forc__1c__1=ⅇaa22b2ⅇaSubstitutec__1=ⅇaa22b2ⅇainto general solution and simplifyyx=ⅇx2ⅇaba2+x22Solution to the IVPyx=ⅇx2ⅇaba2+x22

(6)

See Also

diff

Int

Student

Student[ODEs]

Student[ODEs][ODESteps]