first order - separable

First Order - Separable

A first order separable differential equation is of the form.

{% \frac{dy}{dx} = g(x)h(y) %}

Dividing by {% h(y) %}, we get

{% p(y) \frac{dy}{dx} = g(x) %}

This equation can be integrated as follows

{% \int p(y) dy = \int g(x)dx %}

If {% P(y) %} is the anti-derivative of {% p(y) %} and {% G(x) %} is the anti-derivative of {% g(x) %}, then we have

{% P(y) = G(x) + C %}

{% \frac{dy}{dx} = \frac{3x^2}{y} %}

{% ydy = 3x^2 dx %}

{% \int ydy = \int 3x^2 dx %}

{% \frac{y^2}{2} = x^3 + C %}