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diamong [38]
2 years ago
9

Problem 1: Use the Laplace Transforms to solve:

Mathematics
1 answer:
Leokris [45]2 years ago
3 0

Transforming the ODE yields

L\left\{y'' - y'\right\} = L\left\{e^{-3t}\right\}

(s^2 Y(s) - sy(0) - y'(0)) - (s Y(s) - y(0)) = \dfrac1{s+3}

(s^2 - s) Y(s) = \dfrac1{s+3}

Y(s) = \dfrac1{(s^2 - s)(s+3)} = \dfrac1{s(s-1)(s+3)}

Partial fractions:

Y(s) = \dfrac as + \dfrac b{s-1} + \dfrac c{s+3}

\implies 1 = a(s-1)(s+3) + b s(s+3) + c s(s - 1)

\implies 1 = -3 a + (2 a + 3 b - c) s + (a + b + c) s^2

\implies \begin{cases}-3a=1 \\ 2a+3b-c=0 \\ a+b+c=0\end{cases} \implies a=-\dfrac13, b=\dfrac14, c=\dfrac1{12}

\implies Y(s) = -\dfrac13 \times \dfrac1s + \dfrac14 \times \dfrac1{s-1} + \dfrac1{12} \times \dfrac1{s+3}

Take the inverse transform:

\boxed{y(t) = -\dfrac13 + \dfrac{e^t}4 + \dfrac{e^{-3t}}{12}}

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