Use Laplace transforms to solve the following initial value problem y"-y'-6y=0 y(0)=1 y'(0)=-1

facas9

facas9

Answered question

2021-02-14

Use Laplace transforms to solve the following initial value problem
y"y6y=0
y(0)=1
y(0)=1

Answer & Explanation

Maciej Morrow

Maciej Morrow

Skilled2021-02-15Added 98 answers

Step 1
The given IVP is as follows.
y"y6y=0
y(0)=1
y(0)=1
Apply Laplace transform on the IVP as follows.
yy6y=0
L{y}L{y}L{6y}=L{0}
s2L{y}sy(0)y(0)sL{y}+y(0)6L{y}=0
L{y}[s2s6]=s(1)+(1)(1)
L{y}[s2s6]=s2
L{y}=s2s2s6
L{y}=s2(s3)(s+2)
Step 2
Apply partial fraction on s2(s3)(s+2) as follows.
s2(s3)(s+2)=As3+Bs+2
s2=A(s+2)+B(s3)
s2=(A+B)s+(2A3B)
A+B=1,2A3B=2
A=1B
2(1B)3B=2
5B=4
B=45
A=15
s2(s3)(s+2)=15(1s3)+45(1s+2)
Obtain the solution of the IVP as follows.
L{y}=15(1s3)+45(1s+2)
Take inverse Laplace transforms as follows.
L1{L{y}}=L1{15(1s3)+45(1s+2}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?