Question

Find the solution of the given differential equation and then find the particular solution for which a point (x,y) is given:

dy/dx = x

Answer 1

$$\dfrac{dy}{dx}=x$$

$$dy=xdx$$

Integrate

$$\int dy=\int xdx$$

$$y=\dfrac{1}{2}x^2+C$$

A point $(x_0, y_0)$ is given.

$$y_0=\dfrac{1}{2}x_0^2+C=>C=y_0-\dfrac{1}{2}x_0^2$$

 The particular solution is

$$y=y_0+\dfrac{1}{2}x^2-\dfrac{1}{2}x_0^2$$