Answer:
This differential equation can be solved with the method of the undetermined coefficients
The general solution is yg = yh + yp
where yh is the solution to the homogeneous equation and yp is a particular solution
first of all, we calculate the homogeneous solution
y'' + 36y = 0
with the characteristic polynomial
m = ± 6i
here we have imaginary solutions, hence:
yh = C1*cos(6t) + C2*sin(6t)
With C1 and C2 constants that we have to compute with the initial conditions. For the calculation of yp we assume that one solution is yp = C3*e^(-t)
by calculating the derivatives and replacing:
yp' = -C3*e^(-t)
yp'' = C3*e^(-t)
yp'' + 36yp = e^(-t)
C3*e^(-t) + 36*C3*e^(-t) = e^(-t)
37C3 = 1 --> C3 = 1/37
Hence, the solution is
y = yg = yh + yp = C1*cos(6t) + C2*sin(6t) + (1/37)e^(-t)
By using the initial condition we have
y(0) = C1*cos(0) + C2*sin(0) + (1/37)e^(0) = C1 + 1/37 = 0 --> C1 = -1/37
y'(0) = -C1*sin(0) + C2*cos(0) - (1/37)e^(0) = C2 - 1/37 = 0 --> C2= 1/37
Thus, finally
y = (1/37)( cos(6t) + sin(6t) + e^(-t) )