solve the initial value problem y"+y'-2y=-4;   y(0)  =  y'(0)=0 using Laplace transformsy"+y'-2y=-4;    y(0)  =  y'(0)=0

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lfryerda eNotes educator| Certified Educator

To solve the IVP `y''+y'-2y=-4` with `y(0)=y'(0)=0`, we need to use the following Laplace transforms:





Applying the Laplace transform to the IVP gives the algebraic equation




Now the RHS needs to be separated using partial fractions and comparing coefficients


Comparing coefficients gives `A=2`, `B=-2/3` and `C=-4/3`.

Finally, apply the Inverse Laplace transform to Y to get


`y(t)=2-2/3 e^{-2t}-4/3 e^t`

The solution to the IVP using Laplace transforms is `y(t)=2-2/3 e^{-2t}-4/3e^t`.