A homogeneous differential equation of the from `(dx)/(dy)=h(x/y)`can be solved by making the substitution.(A) `y = v x` (B) `v = y x` (C) `x = v y` (D) `x = v`

Options

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

Answer

C

Explanation

Since the given differential equation `(dx)/(dy)=h((x)/(y))` is homogenous, therefore put `x=vy`,
`(dx)/(dy)=v+y(dv)/(dy)`
`:. V+y(dv)/(dy)=hvimpliesy(dv)/(dy)=v(h-1)`
`implies (1)/((h-1)v)dv=(1)/(y)dy`
On integration, `(1)/((h-1))int(1)/(v)dv=int(dy)/(y)`
`implies (1)/((h-1))log|v|=log|y|+C`