For the differential equation x (dy/dx)+2y=xy (dy/dx),

Question

For the differential equation x (dy/dx)+2y=xy (dy/dx), (A) order is 1 and degree is 1 (B) solution is ln(yx2) = C - y (C) order is 1 and degree is 2 (D) solution is ln(xy2) = C + y

Tardigrade Admin · Accepted Answer

Given, x (dy/dx)(1-y)+2y=0 ⇒ ((1-y/y))dy+2 (dx/x)=0 ⇒ ln y-y+2 ln x=C ⇒ ln(yx2)=C+y

Q. For the differential equation $x \frac{d y}{d x} + 2 y = x y \frac{d y}{d x}$ ,

order is $1$ and degree is $1$

solution is $l n (y x^{2}) = C - y$

order is $1$ and degree is $2$

solution is $l n (x y^{2}) = C + y$

Given, $x \frac{d y}{d x} (1 - y) + 2 y = 0$
$\Rightarrow (\frac{1 - y}{y}) d y + 2 \frac{d x}{x} = 0$
$\Rightarrow l n y - y + 2 l n x = C$
$\Rightarrow l n (y x^{2}) = C + y$

Q. For the differential equation xdxdy​+2y=xydxdy​,

order is 1 and degree is 1

solution is ln(yx2)=C−y

order is 1 and degree is 2

solution is ln(xy2)=C+y