Question

Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`

Answer 1

xdy – ydx = `sqrt(x^2 + y^2)dx`

It is a Homogeneous Equation as

`(dy)/(dx) = (sqrt(x^2 + y^2) + y)/x`

= `sqrt(1 + (y/x)^2) + y/x`

= `f(y/x)`

Put y = vx

`(dy)/(dx) = v + x (dv)/(dx)`

`v + x (dv)/(dx) = sqrt(1 + v^2) + v`

Separating variables, we get

`(dv)/sqrt(1 + v^2) = (dx)/x`

Integrating, we get `log|v + sqrt(1 + v^2)|`

= `log|x| + logK, K > 0`

`log|y + sqrt(x^2 + y^2)| = logx^2K`

⇒ `y + sqrt(x^2 + y^2) = +- Kx^2`

⇒ `y + sqrt(x^2 + y^2) = Cx^2`, which is the required general solution