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

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

2021-01-17
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 , $$L(y'(t))=sL(y(t))-f(0)$$
apply Laplace tranform to
$$y'+4y=e^(-4t) , y(0)=5$$ , to get
$$sF(s)+4F(s)-5=L(e^{-4t})=\frac{1}{(s+4)}$$
Step 3
Find the solution y(t) by taking the inverse Laplace transform of F(s), usimg standard formula
$$F(s)(s+4)=5+\frac{1}{(s+4)}$$
$$F(s)=\frac{5}{(s+4)}+\frac{1}{(s+4)^2}$$
$$\Rightarrow y(t)=L^{-1}(F(s))$$
$$\text{So, } y(t)=5e^{-4t}+te^{-4t}$$
Step 4
ANSWER: $$y(t) = 5e^{-4t}+te^{-4t}$$
Check:
$$y(t)=5e^{-4t}+te^{-4t}$$
$$y'(t)=-20e^{-4t}+e^{-4t}-4te^{-4t}$$
$$\text{so , } y'+4y=e^{-4t} , y(0)=5$$

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