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:

\[\boxed{x^2(1-y)}\] dy + \[\boxed{y^2(1+x)}\] dx = 0 

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

\[\boxed{y^{-2}}\] `dy - 1/y  dy + x^(-2)` dx + \[\boxed{\frac{1}{x}}\]dx = 0

Integrating, we get

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

 ∴ `y^(-1)/(-1)` - \[\boxed{logy}\] + `x^(-1)/(-1)` + \[\boxed{logx}\] = c

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

log x - log y = \[\boxed{\frac{1}{x}+\frac{1}{y}}\]

is the required solution.