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ODE Laplace transforms1

ODE (initial value problems) using Laplace transforms

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ODE Laplace transforms1

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  1. Using Laplace Transforms for solving ODE’s For problems of this type (Paper VII) we have two important results: L(y’’)=s2F(s)-sy(0)-y’(0) L(y’)=sF(s)-y(0)

  2. Example 9y’’-6y’+y=0, Given y(0) =3 and y’(0)=1 Hence 9L[y’’]-6L[y’]+L[y]=0 IMPLIES : 9[s2F(s)-sy(0)-y’(0)]-6[sF(s)-y(0)]+F(s)=0 But Also given y(0) =3 and y’(0)=1 We have 9[s2F(s)-3s-1]-6[sF(s)-3]+F(s)=0 Or [9s2-6s+1]F(s)=27s-9 Or F(s)=9(3s-1)/[9s2-6s+1] Hence y=L-1[F(s)] =9L-1[(3s-1)/9s2-6s+1]

  3. finally you get y= (1/3)e(1/3t) One more problem: y’’+9y=25e4t, giveny(0) =0, y’(0)=7 Here use the hint: L[RHS]=25L[e4t]=25/(s-4) and finally get expression for F(s) and take inverse Laplace transform to get y=f(t)

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