Question

A homogeneous differential equation of the from $\frac{dx}{dy}=h\left(\frac{x}{y}\right)$ can be solved by making the substitution:

• A. $y=vx$
• B. $v=yx$
• C. $x=vy$
• D. $x=v$

Answer 1

Answer: (C)

$x=vy$

A homogeneous differential equation of the form
$\frac{dx}{dy}=h\left(\frac{x}{y}\right)$ can be solved by substituting
$x=vy$
Then
$\frac{dx}{dy}=v+y\frac{dv}{dy}$
Hence
$\frac{dx}{dy}=h\left(\frac{x}{y}\right)$
$\to v+y\frac{dv}{dx}=h\left(v\right)$
$y\frac{dv}{dy}=h\left(v\right)-v$
$\frac{dv}{h\left(v\right)-v}=\frac{dy}{y}$
Let $f\left(v\right)=h\left(v\right)-v$ Then
$\frac{dv}{f\left(v\right)}=\frac{dy}{y}$
Integrating both sides give us
$\int \frac{dv}{f\left(v\right)}=\int \frac{dy}{y}$
$\int \frac{dv}{f\left(v\right)}=ln\left(y\right)+c$