Question

Solve the following differential equation (x² – yx²) dy + (y² + xy²) dx = 0.

Separating the variables, the given equation can be written as:

`square  dy + square  dx = 0` 

∴`(y^(-2) - 1/y)dy + (x^(-2) + 1/x)dx = 0`

`square  dy - 1/y  dy + x^(-2) dx + square  dx = 0`

Integrating, we get

`inty^(-2) dy - int1/y dy + int x^(-2)dx + int 1/x dx = 0`

 ∴ `y^(-1)/(-1) - square + x^(-1)/(-1) + square = c`

`-1/y - 1/x + log x - log y = c`

`log x - log y = square + c`

is the required solution.

Answer 1

Separating the variables, the given equation can be written as:

`bb(x^2(1-y)) dy + bb(y^2(1 + x)) dx = 0` 

∴`(y^(-2) - 1/y)dy + (x^(-2) + 1/x)dx = 0`

`bb(y^(-2))  dy - 1/y  dy + x^(-2)  dx + bb(1/x)  dx = 0`

Integrating, we get

`inty^(-2) dy - int1/y dy + int x^(-2)dx + int 1/x dx = 0`

 ∴ `y^(-1)/(-1) - bb(log y) + x^(-1)/(-1) + bb(log x) = c`

`-1/y - 1/x + log x - log y = c`

`log x - log y = bb((x/y)) + c`

is the required solution.