Question

For the differential equation, find the general solution:

y dx + (x – y²) dy = 0

Answer 1

y dx + (x – y²) dy = 0

or `dx/dy + x/y = y`

This is a linear differential equation of the form `dy/dx + Py = Q.`

Here P = `1/y, Q = y`

∴ `I.F. = e^(int P dx) = e^(int (1/y)dy) = e^(log y) = y`

Hence, the general solution of the differential equation

`x × I.F. = int Q xx (I.F.) dy + C`

⇒ `x xx y = int y xx y  dy + C`

⇒ `xy = int y^2 dy + C` 

⇒ `xy = 1/3 y^3 + C`

⇒ `x = y^2/3 + C/y`

Which is the required solution.