Question

After using \[y=vx\] and writing the right-hand side as \[g(v)\], which separable form is obtained?

Options

• \[\frac{dv}{g(v)+v}=\frac{dx}{x}\]

• \[\frac{dx}{g(v)-v}=\frac{dv}{v}\]

• \[\frac{dv}{g(v)-v}=\frac{dx}{x}\]

• \[\frac{dv}{g(v)-v}=\frac{dy}{y}\]

Answer 1

Substitution gives \[v+x\frac{dv}{dx}=g(v)\]. Thus \[x\frac{dv}{dx}=g(v)-v\], which separates as \[\frac{dv}{g(v)-v}=\frac{dx}{x}\].