Solve the following initial value problem using the method o

Solve the following initial value problem using the method of Laplace transforms: y\" - 2y\' + by = 0, y(0) = 2, y\'(0) = 4.

Solution

L(y\')=sY-y(0)=sY-2

L(y\'\')=s^2Y-sy(0)-y\'(0)=s^2Y-2s-4

Substituting in ODE gives:

s^2Y-2s-4-2(sY-2)+5Y=0

(s^2-2s+5)Y-2s-4+4=0

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

Y=2s/((s^2-(1+4i)s-(1-4i)s+5)

Y=2s/((s-(1+4i))(s-(1-4i)))

Y=1/(s-(1-4i))+(s+1+4i)/((s-(1+4i))(s-(1-4i)))

Y=1/(s-(1-4i))+(s-(1-4i)+(1-4i)+1+4i)((s-(1+4i))(s-(1-4i)))

Y=1/(s-(1-4i))+1/(s-(1+4i))+2/((s-(1+4i))(s-(1-4i)))

Y=1/(s-(1-4i))+1/(s-(1+4i))+(1/4i)(1/(s-(1-4i))-1/(s-(1+4i))

Taking inverse laplace transform:

y(t)=e^{(-1-4i)t}+e^{(1+4i)t}+(1/4i){e^{(-1-4i)t}-e^{(1+4i)t}}