Question

A homogeneous equation of the form $\frac{dy}{dx}=h\left(\frac{x}{y}\right)$ can be solved making the substitution
(a) y=vx
(b) v=yx
(c) x=vy
(d) x=v

Answer 1

Since, given equation $\frac{dy}{dx}=h\left(\frac{x}{y}\right)$ is a homogeneous, so by the substitution $\frac{x}{y}=vi.e.,$ $x=vy\frac{dx}{dy}=v+y\frac{dv}{dy}$
It becomes $v+y\frac{dv}{dy}=hv\Rightarrow y\frac{dv}{dy}=v(h-1)\Rightarrow \frac{1}{(h-1)v}dv=\frac{1}{y}dv$
On integrating both sides, we get
$\frac{1}{(h-1)}\int \frac{1}{v}dv=\int \frac{dy}{y}\Rightarrow \frac{1}{(h-1)}log|v|=log|y|+C$
Hence, (c) is the correct option.