Use-Laplace-transform-to-solve-the-initial-value-problem-ty-4t-2-y-13t-4-y-0-where-y-0-0-and-y-0-0-

Question Number 116457 by bobhans last updated on 04/Oct/20

Use Laplace transform to solve the

initial value problem {ty}^{″} + (4 t - 2) y^{'} + (13 t - 4) y = 0

where y (0) = 0 and y^{'} (0) = 0

Answered by Olaf last updated on 04/Oct/20

L (y^{″}) = p^{2} L (p) - p y (0) - y^{'} (0)

L (y^{″}) = p^{2} L (p)

L (y^{'}) = pL (p) - y (0)

L (y^{'}) = pL (p)

L (t^{n} f (t)) = {(- 1)}^{n} \frac{d^{n} L (p)}{d p^{n}}

L (t f (t)) = - L^{'} (p)

- {(p^{2} L)}^{'} - 4 {(pL)}^{'} - 2 pL - {13 L}^{'} - 4 L = 0

p^{2} L^{'} + 2 pL + {4 pL}^{'} + 4 L + 2 (p + 2) L + {13 L}^{'} = 0

(p^{2} + 4 p + 13) L^{'} + 4 (p + 1) L = 0

\frac{L^{'}}{L} = - 4 \frac{p + 1}{p^{2} + 4 p + 13}

\Rightarrow integation and inverse Laplace

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