Homogeneous equations

Definition

A first order ODE is said to be homogeneous if it can be written in the form of

\frac{dy}{dt} = h\left( \frac{y}{t} \right)

for some function $h$.

Test for homogeneity

\frac{dy}{dt} = h\left( \frac{y}{t} \right) \implies f(\lambda t, \lambda y) = h\left( \frac{\lambda y}{\lambda t} \right) = h\left( \frac{y}{t} \right) = f(t, y)

Let solve such a equation by letting $v=y/t$. Observe that $y=vt$.

$$\begin{array}{rl}\frac{dy}{dt}& =h\left(\frac{y}{t}\right)\\ \frac{d(vt)}{dt}& =h(v)\\ v+t\frac{dv}{dt}& =h(v)\\ \frac{dv}{dt}& =\frac{h(v)-v}{t}\end{array}$$

Now notice that this is a separable equation with respect to $v$.

$$\begin{array}{rl}\frac{dv}{dt}& =\frac{\left(\frac{1}{t}\right)}{\frac{1}{h(v)-v}}\\ ⟹\int \frac{1}{h(v)-v}dv& =\int \frac{1}{t}dt\end{array}$$

One should solve this integrals and finally replace $v$ with $y/t$.