First-order homogeneous equation

Here «homogeneous» does not mean the same as in linear equations with zero right side. It means the equation does not change when \(x\) and \(y\) are multiplied by the same factor: the slope at a point depends only on its direction from the origin. So the direction field is identical along every line through the origin.

That observation suggests the change of variable. Write \(y=vx\), with \(v\) a function of \(x\); by the product rule \(y'=v+xv'\). Substituting in \(y'=F(y/x)\) gives \(xv'=F(v)-v\), a separable equation: \(\frac{dv}{F(v)-v}=\frac{dx}{x}\). Integrate, and at the end return with \(v=y/x\).

Guiding example: in \(y'=\frac{x+y}{x}=1+\frac yx\) we have \(F(v)=1+v\), so \(xv'=1\) and \(v=\ln|x|+C\), hence \(y=x(\ln|x|+C)\). To recognize the type in an equation \(M\,dx+N\,dy=0\), check that \(M\) and \(N\) are homogeneous of the same degree; then divide everything by the corresponding power of \(x\).

If \(F(c)=c\), the line \(y=cx\) is a solution and is lost when dividing by \(F(v)-v\); check it separately. A typical mistake is forgetting to multiply by \(x\) when returning to \(y\). This is the first substitution in the course; the Bernoulli equation will use a similar idea to reach a linear equation.