The solution of the differential equation, (d x/d y)=(y/2 y ℓ n y+y-x) is

Question

The solution of the differential equation, (d x/d y)=(y/2 y ℓ n y+y-x) is (A) x y=y3 ℓ n y+c (B)  x y=y2 ln y +c (C) x y=y5 ℓ n y+c (D) x y=y4 ℓ ny +c

Tardigrade Admin · Accepted Answer

(d y/d x)=(y/2 y ℓ n y+y-x) ⇒ (d x/d y)+(x/y)=(2 ℓ n y+1) I.F. e∫ (1/y) d y=e ln y=y solution is y x=∫(2 ℓ ny +1) y d y+c x y=y2 text lny +c

Q. The solution of the differential equation, $\frac{d x}{d y} = \frac{y}{2 y ℓ n y + y - x}$ is

$x y = y^{3} ℓ n y + c$

$x y = y^{2} ln y + c$

$x y = y^{5} ℓ n y + c$

$x y = y^{4} ℓ n y + c$

$\frac{d y}{d x} = \frac{y}{2 y ℓ n y + y - x}$
$\Rightarrow \frac{d x}{d y} + \frac{x}{y} = (2 ℓ n y + 1)$
I.F. $e^{\int \frac{1}{y} d y} = e^{l n y} = y$
solution is $y x = \int (2 ℓ$ ny $+ 1) y d y + c$
$x y = y^{2} lny + c$

Q. The solution of the differential equation, dydx​=2yℓny+y−xy​ is

xy=y3ℓny+c

xy=y2lny+c

xy=y5ℓny+c

xy=y4ℓny+c