Using partial fractions method solve the initial value probl

Using partial fractions method solve the initial value problem y\"-y\'-6y=0 given y(0)=6 and y’(0)=13.

Please show work. Thanks!

Solution

We take laplace transform

L(y\')=sY-y(0)=sY-6

L(y\'\')=s^2Y-sy(0)-y\'(0)=s^2Y-6s-13

Substituting gives

s^2Y-6s-13-sY+6-6Y=0

(s^2-s-6)Y=6s+7

(s-3)(s+2)Y=6s+7

Let,

6s+7=a(s-3)+b(s+2)

So,

6=a+b

7=-3a+2b

Solving gives, b=5,a=1

So,

(s-3)(s+2)Y=(s-3)+5(s+2)

Y=1/(s+2)+5/(s-3)

Taking inverse laplace tranform gives

y=e^{-2t}+5e^{3t}