Use Laplace transforms to solve the following initial value problem x"-25x=9t x(0)=x'(0)=0

Lennie Carroll 2021-01-19 Answered
Use Laplace transforms to solve the following initial value problem
x"25x=9t
x(0)=x(0)=0
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Expert Answer

Nichole Watt
Answered 2021-01-20 Author has 100 answers

Step 1
Given:
Given IVP is x"25x=9t,x(0)=x(0)=0
To find solution of IVP we use Laplace transform
step 2
Solution:
Consider,
x"25x=9t
Apply Laplace transform on both side we get
L{x25x}=L{9t}
L{x}25L{x}=9L{t}
(s2X(s)sx(0)x(0))25X(s)=9s2
Given initial conditions are x(0)=x(0)=0 therefore we get
(s2X(s)s(0)0)25X(s)=9s2
(s225)X(s)=9s2
X(s)=9s2(s225)
Now we use inverse Laplace transform to find X(t)
X(t)=L1{9s2(s225)}
X(t)=L1{9s2(s+5)(s5)}(1)
Now we take partial fraction
9s2(s+5)(s5)=As2+B(s+5)+C(s5)
A(s+5)(s5)+Bs2(s5)+Cs2(s+5)=9
A(s225)+B(s35s2)+C(s3+5s2)=9
(B+C)s3+(A5B+5C)s225A=9
comparing both side we get system of equations,
{25A=9A=925A5B+5C=05B+5C=925(2)B+C=0(3)
5B+5C=925
+5B+5C=0
10C=925
Therefore C=9250 substituting value of C in equation 3 we get
B+C=0B=CB=9250
Therefore Partial fraction of 9s2(s+5)(s5)=9(25s2)9250(s+5)+9250(s5) Hence 1 becomes,
X(t)=L1{9s2(s+5)(s5)}
=L1{925s29250(s+5)+9250(s5)}

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