Simplifying First Order ODEs via substitution

Certain ODEs can be simplified into basic linear ODEs or separable ODEs via substitution.

Homogenous Type First Order ODEs §

A first order homogenous ODE is one of the form: $\frac{dy}{dx}=f(\frac{y}{x})$ First order homogenous ODEs can be simplified by using a substitution of $u=\frac{y}{x}$ . Note that since a change of variable has occurred, we need to derive $\frac{du}{dx}$ (via the Product rule).

Warning not the same as as a homogenous ODE in higher power ODEs.

Note that this is

1: Homogenous ODE

Solve the following ODE: $\frac{dy}{dx}=\frac{y}{x}+{\cos }^2(\frac{y}{x})$

Let $u=\frac{y}{x}⟹y=ux$ Then $\frac{dy}{dx}=\frac{d}{dx}(ux)=\frac{du}{dx}x+u$ (Product Rule) Subbing back into original equation: $\frac{du}{dx}x+\cancel{u}=\cancel{u}+{\cos }^2(u)$ $\frac{du}{dx}x={\cos }^2(u)⟹\frac{1}{{\cos }^2(u)}\frac{du}{dx}=x$ $\int sec(u{)}^{2}du=\int xdx$ $\tan (u)=\frac{x^2}{2}+c$ $\frac{y}{x}=\arctan (\frac{x^2}{2}+c)$ $y=x\arctan (\frac{x^2}{2}+c)$

Bernoulli’s Equation §

#todo Which equation exactly?

The equation has the form: $\frac{dy}{dx}+P(x)y=Q(x)y^n$ By substituting $u=y^{1-n}$, the equation can be reduced to a linear ODE