Question

Show that y² − x² − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]

Answer 1

We have,

\[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0\]

Now,y² − x² − xy = a

`rArr2y(dy)/(dx)-2x-y-x(dy)/(dx)=0`

`rArr(2y-x)(dy)/(dx)-2x-y=0`

`rArr(2y-x)(dy)/(dx)=2x+y`

`rArr(x-2y)(dy)/(dx)=-(2x+y)`

`rArr(x-2y)(dy)/(dx)+2x+y=0`

Thus, y² − x² − xy = a is the solution of the given differential equation.