152. Find the particular solution of the equation $y^{\prime \prime}-2 y^{\prime}+50 y=0$, if $y=1$ and $y^{\prime}=1$ when $x=0$.
Solution. The characteristic equation $k^{2}-2 k+50=0$ has roots $k_{1,2}=1 \pm \sqrt{1-50}=1 \pm 7 i$. Therefore, the general solution of the equation is

$$
y=e^{x}\left(C_{1} \cos 7 x+C_{2} \sin 7 x\right)
$$

Next, we find

$$
y^{\prime}=e^{x}\left(C_{1} \cos 7 x+C_{2} \sin 7 x\right)+e^{x}\left(-7 C_{1} \sin 7 x+7 C_{2} \cos 7 x\right)
$$

Substituting $x=0, y=1$ into the expression for $y$, we get $C_{1}=1$. Then, substituting $x=0, y^{\prime}=-1, C_{1}=1$ into the expression for $y^{\prime}$, we find $C_{2}=$ $=-2 / 7$.

Thus, the particular solution is

$$
y=e^{x}\left(\cos 7 x-\frac{2}{7} \sin 7 x\right)
$$