Question

Verify implicit solution 2xydx+(x^2-y)dy=0; -2x^2y+y^2)=1

Answer 1

If
`-2x^2y+y^2=1`, taking the derivative of both sides wrt
`x` gives


`-4xy-2x^2(\mathrm dy)/(\mathrm dx)+2y(\mathrm dy)/(\mathrm dx)=0`

Solving for
`(\mathrm dy)/(\mathrm dx)` gives


`-4xy+(-2x^2+2y)(\mathrm dy)/(\mathrm dx)=0`

`(-2x^2+2y)(\mathrm dy)/(\mathrm dx)=4xy`

`(-2x^2+2y)\mathrm dy=4xy\,\mathrm dx`

`(-x^2+y)\mathrm dy=2xy\,\mathrm dx`

`0=2xy\,\mathrm dx+(x^2-y)\mathrm dy`