Example 4.14. Find the general solution of the equation

$$
y^{\prime \prime}=1-x^{2}
$$
Solution. Let $\frac{d y}{d x}=z$, then the given equation can be written as:

$$
\frac{d z}{d x}=1-x^{2}, \text { or } d z=\left(1-x^{2}\right) d x
$$

After integration, we find

$$
z=x-\frac{x^{3}}{3}+C_{1}
$$

Hence,

$$
\frac{d y}{d x}=x-\frac{x^{3}}{3}+C_{1}, \text { or } \quad d y=x-\frac{x^{3}}{3}+C_{1} d x .
$$

Integrating this equation, we obtain the general solution

$$
y=\frac{x^{2}}{2}-\frac{x^{4}}{12}+C_{1} x+C_{2} .
$$