1357. Check that the function $y=x^{2}+C$ is a solution to the differential equation $\frac{d y}{d x}=2 x$. Construct the family of integral curves and highlight the integral curve passing through the point $(1 ; 2)$.
Solution. Let's find the derivative $\frac{d y}{d x}=\frac{d}{d x}\left(x^{2}+C\right)=2 x$ and substitute it into the given equation. We obtain the identity

$$
2 x=2 x
$$

Therefore, $y=x^{2}+C$ is a solution to the given differential equation and, moreover, a general solution since it contains an arbitrary constant. The equation

$$
y=x^{2}+C
$$

defines a family of parabolas (Fig. 97), which will be the family of integral curves. Let's find the integral curve (parabola) passing through the given point ( $1 ; 2$ ). Substituting the coordinates of this point into equation (a), we get $2=1^{2}+C$, from which $C=1$, hence the function $y=x^{2}+1$ is a particular solution to the given differential equation, and its graph (integral curve) will pass through the given point.

Verify that the specified functions are solutions to the differential equations:

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-271.jpg?height=398&width=325&top_left_y=289&top_left_x=777)

Fig. 97