Solve the differential equation using Laplace transform of y''-3y'+2y=e^{3t} when y(0)=0 and y'(0)=0

glasskerfu 2021-02-19 Answered
Solve the differential equation using Laplace transform of
y3y+2y=e3t
when y(0)=0 and y'(0)=0
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Expert Answer

Khribechy
Answered 2021-02-20 Author has 100 answers
Step 1
To Solve:
y3y+2y=e3t
when y(0)=0 and y'(0)=0
Concept:
Let L(y(t))=Y(s) be any given Laplace Transform
Then laplace transform of the heigher order derivative is given by the following equation.
L(y)=(sY(s)y(0))
L(y)=(s2Y(s)sy(0)y(0))
L(eat)=1(sa)
Step 2
Explanation:
We havey3y+2y=e3t
taking the laplace transform of each term, we get
L{y}3L{y}+2L{y}=L{e3t}
s2Y(s)sy(0)y(0)3sY(s)y(0)+2Y(s)=1(s3)
s2Y(s)3sY(s)+2Y(s)=1(s3)(y(0)=y(0)=0)
Y(s)[s23s+2]=1(s3)[s23s+2]
Y(s)=12(s1)1(s2)+12(s3)
Now, taking the inverse of the laplace, we get
y(t)=L1{Y(s)}=12L1{1(s1)}L1{1(s2)}+12L1{1(s3)}
y(t)=12ete2t+12e3t
Therefore, the solution of the given Initial value problem is
y(t)=12ete2t+12e3t
Step 3
Answer:
y(t)=12ete2t+12e3t
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