Consider the initial value problem,
$y^{\prime \prime}+2 y^{\prime}+5 y=4 e^{-t} \cos 2 t, y(0)=1,$ $y^{\prime}(0)=0$
This is a non-homogeneous differential equation. Then its general solution is given by, $y=$complimentary solution $+$ particular solution.
For the complimentary solution, consider the homogeneous equation
$y^{\prime \prime}+2 y^{\prime}+5 y=0$
Then its characteristic equation is given by,
$r^{2}+2 r+5=0$
Then,
$r=\frac{-2 \pm \sqrt{4-20}}{2}$
$r=\frac{-2 \pm 4 i}{2}$
$r=-1 \pm 2 i$
Therefore, the complimentary solution is,
$y_{c}(t)=c_{1} e^{-t} \cos 2 t+c_{2} e^{-t} \sin 2 t$
where $c_1$ and $c_2$ are arbitrary constant.
For the particular solution, by the method of undetermined co-efficients, suppose $Y(t)=A t e^{-t} \cos 2 t+B t e^{-t} \sin 2 t$ is a function which satisfies the equation $Y^{\prime \prime}+2 Y^{\prime}+5 Y=4 e^{-t} \cos 2 t$
Since $Y=A t e^{-t} \cos 2 t+B t e^{-t} \sin 2 t$
Then $Y^{\prime}=A e^{-t} \cos 2 t-A t e^{-t} \cos 2 t-2 A t e^{-t} \sin 2 t+B e^{-t} \sin 2 t -B t e^{-t} \sin 2 t+2 B t e^{-t} \cos 2 t$
$=(A-A t+2 B t) e^{-t} \cos 2 t+(-2 A t+B-B t) e^{-t} \sin 2 t$
And
$Y^{\prime \prime}=(2 A-3 A t+4 B-4 B t) e^{-t} \cos 2 t+(-4 A+4 A t-2 B-3 B t) e^{-t} \sin 2 t$
Substitute $Y,Y'$ and $Y''$ in $Y^{\prime \prime}+2 Y^{\prime}+5 Y=4 e^{-t} \cos 2 t$ to get,
$(-2 A-3 A t+4 B-4 B t+2 A-2 A t+4 B t+5 A t) e^{-t} \cos 2 t +(-4 A+4 A t-2 B-3 B t-4 A t+2 B-2 B t+5 B t) e^{-t} \sin 2 t=4 e^{-t} \cos 2 t$
$4 B e^{-t} \cos 2 t-4 A \sin 2 t=4 e^{-t} \cos 2 t$
$4 B \cos 2 t-4 A \sin 2 t=4 \cos 2 t$
On equating coefficients on both sides,
$B=1$ and $A=0$
Then, the particular solution is,
$Y(t)=t e^{-t} \sin 2 t$
So, the general solution of $y^{\prime \prime}+2 y^{\prime}+5 y=4 e^{-t} \cos 2 t, y(0)=1,$ $y^{\prime}(0)=0$ is,
$y=y_{c}(t)+Y(t)$
$y=c_{1} e^{-t} \cos 2 t+c_{2} e^{-t} \sin 2 t+t e^{-t} \sin t$
Differentiate this equation with respectt to $t$, to get,
$y^{\prime}=-c_{1} e^{-t} \cos 2 t-2 c_{1} e^{-t} \sin 2 t-c_{2} e^{-t} \sin 2 t+2 c_{2} e^{-t} \cos 2 t +e^{-t} \sin 2 t-t e^{-t} \sin 2 t+2 t e^{-t} \cos 2 t$
Use the initial condition $y(0)=1$, implies,
$1=c_1+0+0$
$c_1 = 1$
Use the initial condition $y'(0)=0$, implies,
$0=-c_{1}+2 c_{2}$
$c_{2}=\frac{1}{2}$
Hence the required solution of given initial value problem is,
$y=e^{-t} \cos 2 t+\frac{1}{2} e^{-t} \sin 2 t+t e^{-t} \sin 2 t$
