Use the Laplace transform to solve the given initial-value problem. y'+4y=e-4t, y(0)=5

Question

Use the Laplace transform to solve the given initial-value problem.

y^{'} + 4 y = e - 4 t, y (0) = 5

Answer 1

Step 1
To apply Laplace transform technique to obtain the solution of the given initial value problem
Step 2
Obtain an equation for

F (s) = L (y (t))

Recall ,

L (y^{'} (t)) = s L (y (t)) - f (0)

apply Laplace tranform to

y^{'} + 4 y = e^{(} - 4 t), y (0) = 5

, to get

s F (s) + 4 F (s) - 5 = L (e^{- 4 t}) = \frac{1}{(s + 4)}

Step 3
Find the solution y(t) by taking the inverse Laplace transform of F(s), usimg standard formula

F (s) (s + 4) = 5 + \frac{1}{(s + 4)}

F (s) = \frac{5}{(s + 4)} + \frac{1}{(s + 4)^{2}}

\Rightarrow y (t) = L^{- 1} (F (s))

So, y (t) = 5 e^{- 4 t} + t e^{- 4 t}

Step 4
ANSWER:

y (t) = 5 e^{- 4 t} + t e^{- 4 t}

Check:

y (t) = 5 e^{- 4 t} + t e^{- 4 t}

y^{'} (t) = - 20 e^{- 4 t} + e^{- 4 t} - 4 t e^{- 4 t}

so , y^{'} + 4 y = e^{- 4 t}, y (0) = 5

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