Question

Find the general solution and indicate method used:-
yy' + xy²= x​

Answer 1

Step-by-step explanation:

We have,

$y \frac{dy}{dx} + x {y}^{2} = x$

$Let y^{2}=t$

$\implies 2y\frac{dy}{dx}=\frac{dt}{dx}$

so,

$\implies \frac{1}{2} \frac{dt}{dx} + x t = x$

$\implies \frac{dt}{dx} +2 x t = 2x$

$I.F. = {e}^{ \int2xdx} = {e}^{ {x}^{2} }$

$\therefore \: t {e}^{ {x}^{2} } = \int \: 2x {e}^{ {x}^{2} } dx$

$\implies \: t {e}^{ {x}^{2} } = {e}^{ {x}^{2} } + c$

$\implies \: {y}^{2} {e}^{ {x}^{2} } = {e}^{ {x}^{2} } + c$