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

First we solve the integral by separating the terms:

$\frac{dy}{dx}={x} \to \intop dy=y=\int{xdx}=\frac{x^{2}}{2}+C \\\implies y=\frac{x^{2}}{2}+C$

Then, we substitute the coordinates (x_i,y_i) to find C for the particular solution:

$y_i=\frac{x_i^{2}}{2}+C \implies C=y_i-\frac{x_i^{2}}{2}$

In conclusion:

$\text{General solution: } \\ y=\frac{x^{2}}{2}+C \\ \text{Particular solution } (x_i,y_i):\\ y=\frac{x^{2}}{2}+y_i-\cfrac{x_i^{2}}{2} ​$