Question

The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:

(where C is a constant of integration)

Options

• y = `Ce^(2(sqrt(x/y))`

• y = `Ce^(-2sqrt(xy))`

• y = `Ce^(2sqrt(xy))`

• y = `Ce^(2(sqrt(y/x))`

Answer 1

`bb(y = Ce^(2(sqrt(x/y)))`

Explanation:

We have, `dy/dx = y/(x + sqrt(xy))`

`\implies dx/dy = (x + sqrt(xy))/y = x/y + sqrt(x/y)`

Put x = vy

`\implies dx/dy = v + y (dv)/dy`

`\implies v + y (dv)/dy = v + sqrt(v)`

`\implies y (dv)/dy = sqrt(v)`

`\implies (dv)/sqrt(v) = dy/y`

`\implies int (dv)/sqrt(v) = int dy/y`

`\implies 2sqrt(v)` = log y + log C'

`\implies 2sqrt(x/y)` = log (C' y)

`\implies` C' y = `e^(2(sqrt(x/y))`

`\implies` y = `1/C^' e^(2(sqrt(x/y))`  ...[∵ 1 / C' = C]

`\implies` y = `Ce^(2(sqrt(x/y))`