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

Tyra 2021-01-16 Answered
Use the Laplace transform to solve the given initial-value problem.
y+4y=e4t,y(0)=5
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Expert Answer

Jaylen Fountain
Answered 2021-01-17 Author has 170 answers
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(e4t)=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+1(s+4)
F(s)=5(s+4)+1(s+4)2
y(t)=L1(F(s))
So, y(t)=5e4t+te4t
Step 4
ANSWER: y(t)=5e4t+te4t
Check:
y(t)=5e4t+te4t
y(t)=20e4t+e4t4te4t
so , y+4y=e4t,y(0)=5
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