# First order differential equations

## Solution by substitution

We often convert complicated differential equations into simple differential equations by substitution. We will deal with two types of substitutions:

**Homogeneous
differential equation,** where substitution either $u=y/x$ or
$u=x/y$ brings DE into simplier one. E.g.

$$
\begin{equation}
\frac{dy}{dx} - \frac{y}{x} - \sin \frac y x = 0 \nonumber \\
(1+ \frac{x^2}{y^2})\ dx + (1-\frac y x)\ dy = 0 \nonumber
\end{equation}
$$
**Bernoulli's
differential equation.** This kind can be converted into linear DE
by eliminating $y^n$ on the right side.

$$
\frac{dy}{dx}+P(x)y = f(x)\cdot \color{red}{y^n} \nonumber
$$

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