1193. Find the first four terms of the power series expansion of the particular integral of the equation $y^{\prime}+x y^{2}=2 \cos x$, satisfying the initial condition: $y(0)=1$.
Solution. As in the previous problems, we seek the integral in the form of a power series (1).

According to the initial condition $y(0)=a_{0}=1$.

Further, finding the series for $y^{2}$ and $y^{\prime}$ and substituting them and the series for $\cos x$

$$
\cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\ldots
$$

into the given equation, we obtain

$$
a_{1}+\left(1+2 a_{2}\right) x+\left(2 a_{1}+3 a_{3}\right) x^{2}+\ldots=2-x^{2}+\ldots
$$

From here, by comparing the coefficients of the same powers of $x$ from both parts of the equality, we find $a_{1}=2 ; a_{2}=-\frac{1}{2}$; $a_{3}=-\frac{5}{3}$.

Therefore, the desired particular integral is

$$
y=1+2 x-\frac{1}{2} x^{2}-\frac{5}{3} x^{3}+\ldots
$$