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