Question

Solve the differential equation: $y^{3}- \frac{dy}{dx}= x^{2} \frac{dy}{dx}$

Answer 1

To solve the differential equation:
$y^{3}- \frac{dy}{dx}= x^{2} \frac{dy}{dx}$

first separate the variables

$y^{3}- \frac{dy}{dx}= x^{2} \frac{dy}{dx} \\ \\ {y}^{3} = {x}^{2} \frac{dy}{dx} + \frac{dy}{dx} \\ \\ {y}^{3} = \frac{dy}{dx} ( {x}^{2} +1) \\ \\ \int\frac{1}{ {x}^{2} +1} dx = \int\frac{1}{ {y}^{3} } dy$
integrate both sides

$tan^{-1}x = - \frac{1}{2 {y}^{2} } + c \\ \\ - 1 + 2c {y}^{2} = 2c {y}^{2}×tan^{-1}x \\ \\2 c {y}^{2} =2c {y}^{2}tan^{-1}x+1$