Solving ODEs With the Laplace Transform in Matlab

3/23/2015 SolvingODEswiththeLaplacetransforminMatlab

http://terpconnect.umd.edu/~petersd/246/matlablaplace.html 1/4

SolvingODEswiththeLaplaceTransforminMatlabThisapproachworksonlyfor

lineardifferentialequationswithconstantcoefficientsrighthandsidefunctionswhicharesumsandproductsof

polynomialsexponentialfunctionssineandcosinefunctionsHeaviside(step)functionsDirac(impulse)``functions''

initialconditionsgivenatt=0

Themainadvantageisthatwecanhandlerighthandsidefunctionswhicharepiecewisedefined,andwhichcontainDiracimpulse``functions''.

Youmustfirstsavethefilemyplot.minyourdirectory.Unfortunately,theezplotfunctionisbuggyinsomeversionsofMatlab.Ifezplotdoesnotwork,trytousemyplotinstead.

Simpleexample

Considertheinitialvalueproblem

y''+3y'+2y=et,y(0)=4,y'(0)=5

Definethenecessarysymbolicvariables:

symsstY

DefinetherighthandsidefunctionandfinditsLaplacetransform:

f=exp(t)F=laplace(f,t,s)

FindtheLaplacetransformofy'(t):Y1=sYy(0)

Y1=s*Y4

FindtheLaplacetransformofy''(t):Y2=sY1y'(0)

Y2=s*Y15

SettheLaplacetransformofthelefthandsideminustherighthandsidetozeroandsolveforY:

Sol=solve(Y2+3*Y1+2*YF,Y)

FindtheinverseLaplacetransformofthesolution:

sol=ilaplace(Sol,s,t)

Examplewithpiecewisedefinedrighthandsidefunction




y''+3y'+2y=f(t),y(0)=2,y'(0)=3

withtherighthandsidefunction

f(t)=1fort




Y2=s*Y13





Plotthesolution:(usemyplotifezplotdoesnotwork)

ezplot(sol,[0,10])

ExamplewithDirac``function''


y''+2y'+10y=1+5 (t5),y(0)=1,y'(0)=2

Definethenecessarysymbolicvariables:

symsstY

Definetherighthandsidefunction:

f=1+5*dirac(t5)

FindtheLaplacetransformoftherighthandsidefunction:

F=laplace(f,t,s)

FindtheLaplacetransformofy'(t):Y1=sYy(0)

Y1=s*Y1


Y2=s*Y12







Plotthesolution:(usemyplotifezplotdoesnotwork)

ezplot(sol,[0,10])

Solving ODEs With the Laplace Transform in Matlab

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