Show that `(3xy + x^(2)) dy + (y^(2) + xy)dx = 0` is homogeneous differential equation. Also find its general solution.

Options

A. `x^(2)(2xy+y^(2))=c^(2)`
B. `x^(2)(2xy-y^(2))=c^(2)`
C. `x^(2)(y^(2)-2xy)=c^(2)`
D. None of these

Answer

A

Explanation

Homogeneous equation can be written in the form of
`(dy)/(dx)=-(3xy+y^(2))/(x^(2)+xy)`
Put `y= v x and (dy)/(dx)=v+x(dv)/(dx)` we, get
`v+xx(dv)/(dx)=(-2v(v+2))/(v+1)`
`rArr (1)/(x)=-((v+1))/(2v(v+2))dv`
`rArr-(2)/(x)=- [ (1)/(2(v+2))+(1)/(2v)]dv`
On integrating, we get
`-2 log e^(x)=(1)/(2) log (v+2)+(1)/(2)log v- log c`
`rArr v(v+2)x^(4)=c^(2)`
`rArr (y)/(x)((y)/(x)+2)x^(4)=c^(2)" " ( :. v=(y)/(x))`
Hence, required solution is `(y^(2)+2xy)x^(2)=c^(2)`