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