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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