Solving ODEs with the Laplace Transform in Matlab This approach works only for linear differential equations with constant coefficients righthand side functions which are sums and products of polynomials exponential functions sine and cosine functions Heaviside (step) functions Dirac (impulse) ``functions'' initial conditions given at t =0 The main advantage is that we can handle righthand side functions which are piecewise defined, and which contain Dirac impulse ``functions''. You must first save the file myplot.m in your directory. Unfortunately, the ezplot function is buggy in some versions of Matlab. If ezplot does not work, try to use myplot instead. Simple example Consider the initial value problem y'' + 3 y'+2 y = e t , y(0) = 4 , y'(0) = 5 Define the necessary symbolic variables: syms s t Y Define the righthand side function and find its Laplace transform: f = exp(‐t) F = laplace(f,t,s) Find the Laplace transform of y'(t): Y 1 = sY y(0) Y1 = s*Y ‐ 4 Find the Laplace transform of y''(t): Y 2 = sY 1 y'(0) Y2 = s*Y1 ‐ 5 Set the Laplace transform of the left hand side minus the right hand side to zero and solve for Y: Sol = solve(Y2 + 3*Y1 + 2*Y ‐ F, Y) Find the inverse Laplace transform of the solution: sol = ilaplace(Sol,s,t) Example with piecewise defined righthand side function

Solving ODEs With the Laplace Transform in Matlab

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Solve Ode's With Laplace Transforms

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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
3/23/2015 SolvingODEswiththeLaplacetransforminMatlab

http://terpconnect.umd.edu/~petersd/246/matlablaplace.html 2/4

Considertheinitialvalueproblem

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

withtherighthandsidefunction

f(t)=1fort
3/23/2015 SolvingODEswiththeLaplacetransforminMatlab

http://terpconnect.umd.edu/~petersd/246/matlablaplace.html 3/4

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

Y2=s*Y13

SettheLaplacetransformofthelefthandsideminustherighthandsidetozeroandsolveforY:

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

FindtheinverseLaplacetransformofthesolution:

sol=ilaplace(Sol,s,t)

Plotthesolution:(usemyplotifezplotdoesnotwork)

ezplot(sol,[0,10])

ExamplewithDirac``function''

Considertheinitialvalueproblem

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

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

Y2=s*Y12
3/23/2015 SolvingODEswiththeLaplacetransforminMatlab

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

SettheLaplacetransformofthelefthandsideminustherighthandsidetozeroandsolveforY:

Sol=solve(Y2+2*Y1+10*YF,Y)

FindtheinverseLaplacetransformofthesolution:

sol=ilaplace(Sol,s,t)

Plotthesolution:(usemyplotifezplotdoesnotwork)

ezplot(sol,[0,10])