1108. Find the particular integral of the equation satisfying the given initial conditions:
1) $y^{\prime \prime}+4 y^{\prime}+5 y=0 ; \quad y(0)=-3, \quad y^{\prime}(0)=0$.
2) $y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0 ; y(0)=-1, y^{\prime}(0)=2, y^{\prime \prime}(0)=3$.
Solution. 1) First, we find the general integral of the given equation. Its characteristic equation $r^{2}+4 r+5=0$ has roots $r_{1,2}=-2 \pm i$. Therefore, according to rule 2, the general integral is $y=e^{-2 x}\left(C_{1} \cos x+C_{2} \sin x\right)$.

Next, using the initial conditions, we determine the values of the constants $C_{1}$ and $C_{2}$. Substituting the given values $x=0, y=-3$ (the first initial condition) into the general integral, we get

$$
-3=e^{0}\left(C_{1} \cos 0+C_{2} \sin 0\right) \text { or }-3=C_{1} \text {. }
$$

Differentiating the general integral (as a product),

$$
y^{\prime}=e^{-2^{x}}\left|\left(C_{2}-2 C_{1}\right) \cos x-\left(C_{1}+2 C_{2}\right) \sin x\right|
$$

and substituting the given values $x=0, y^{\prime}=0$ (the second initial condition) into the result, we get a second equation with the unknowns $C_{1}$ and $C_{2}$:

$$
0=e^{0}\left|\left(C_{2}-2 C_{1}\right) \cos 0-\left(C_{1}+2 C_{2}\right) \sin 0\right| \text { or } C_{2}-2 C_{1}=0 \text {. }
$$

Solving the obtained equations as a system, we find $C_{1}=-3$, $C_{2}=-6$.

Substituting the values of $C_{1}$ and $C_{2}$ into the general integral, we get the desired particular integral of the given equation, satisfying the given initial conditions: $y=-3 e^{-2 x}(\cos x+2 \sin x)$.

2) The characteristic equation $r^{3}+3 r^{2}+3 r+1=0$ or $(r+1)^{3}=0$ of the given differential equation has a triple real root $r=-1\left(r_{1}=r_{2}=r_{3}=-1\right)$. According to rule 3, the general integral is $y=e^{-x}\left(C_{1}+C_{2} x+C_{3} x^{2}\right)$.

Differentiating it twice,

$$
\begin{gathered}
\left.y^{\prime}=e^{-x} i C_{2}-C_{1}+\left(2 C_{3}-C_{2}\right) x-C_{3} x^{2}\right] \\
y^{\prime}=e^{-x}\left[C_{1}-2 C_{2}+2 C_{3}+\left(C_{2}-4 C_{3}\right) x+C_{3} x^{2}\right]
\end{gathered}
$$

and substituting the given values of $y, y^{\prime}$, and $y^{\prime \prime}$ at $x=0$ into the expressions for $y, y^{\prime}$, and $y^{\prime \prime}$, we get a system of three equations for determining the constants $C_{1}, C_{2}, C_{3}$: $-1=C_{1} ; 2=C_{2}-C_{1} ; 3=C_{1}-2 C_{2}+2 C_{3}$, from which we find $C_{1}=-1 ; C_{2}=1 ; C_{3}=3$. Therefore, the desired particular integral is $y=e^{-x}\left(3 x^{2}+x-1\right)$.