Answer to Question #230927 in Differential Equations for syra

Question

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

dy/dx = x

Answer 1

We may rewrite the equation in form

$dy = x\cdot dx,$

next, we integrate both parts and get

$\int dy = \int x\cdot dx, \\ y = \dfrac12\cdot x^2 + c.$

If the point $(x_0, y_0)$ is given, we may obtain the constant c:

$c = y_0 - \dfrac12\cdot x_0^2\,,$

so the solution will be

$y = \dfrac12x^2 + y_0 - \dfrac12x_0^2\,.$