Question

Solve dy÷dx+y÷x=y^2÷x^2

Answer 1

put y=vx

so,

$\frac{dy}{dx} = v + x \frac{dv}{dx}$

therefore,

$v + x \frac{dv}{dx} + v = {v}^{2} \\ \frac{dv}{v.(v - 2 )} = \frac{dx}{x} \\ \frac{1}{2} ( \frac{dv}{v - 2} - \frac{dv}{v} ) = \frac{dx}{x } \\ integrating \: both \: sides \: we \: get \\ log(v - 2) - log(v) = 2 log(x) + log(c) \\ log( \frac{v - 2}{v} ) = log(c( {x}^{2}) ) \\ \\ taking \: antilog \: both \: sides \\ \frac{v - 2}{v} = c {x}^{2} \\ putting \: value \: of \: v \\ \frac{y - 2x}{y} = c {x}^{2}$

---------------That's all-------------------