Besides laplace transform state other two methods that solve the given DE

Question

Solve the initial value problem by using Laplace transform.

y"-9y=3e^(-t) ; y (0) = 0, y'(0)=1

b) Besides Laplace Transform, state two methods that can be used to solve the differential equation.

Answer 1

y"-9y=3e^-t

r^2-9=0

r=3,-3

y_h(t)=c₁e^(3t)+c₂e^(-3t)

find constant using initial values

y=c₁e^(3t)+c₂e^(-3t)

y'=3c₁e^(3t)-3c₂e^(-3t)

0=c₁₊c₂

₁₌3c₁-3c₂

c_{1= 1/6 and} c_2=-1/6

_{find particular solution}

_{(D^2-9)y=3e^-t}

_{y=(3e^-t)/(D^2-9)}

_{y=(3e^-t)/(-1^2-9)}

_y=(-3e^-t)/8

_Solution

y(t)=1/6e^(3t)-1/6e^(-3t)-3/8(e^-t)