If y = y ( x ) is the solution of the differential equation 2 x2 (d y/d x)-2 x y+3 y2=0 such that y(e)=(e/3), then y(1) is equal to

Question

If y = y ( x ) is the solution of the differential equation 2 x2 (d y/d x)-2 x y+3 y2=0 such that y(e)=(e/3), then y(1) is equal to (A) (1/3) (B) (2/3) (C) (3/2) (D) 3

Tardigrade Admin · Accepted Answer

( dy / dx )-( y / x )=-(3/2)(( y / x ))2 y = vx ( dv / v 2)=-(3 dx /2 x ) -(1/ v )=-(3/2) ln | x |+ C -( x / y )=(-3/2) ln | x |+ C x = e , y =( e /3) C =-(3/2) When x =1, y =(2/3)

Q. If $y = y (x)$ is the solution of the differential equation $2 x^{2} \frac{d y}{d x} - 2 x y + 3 y^{2} = 0$ such that $y (e) = \frac{e}{3}$ , then $y (1)$ is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{3}{2}$

$3$

$\frac{d y}{d x} - \frac{y}{x} = - \frac{3}{2} (\frac{y}{x})^{2}$
$y = vx$
$\frac{d v}{v ^{2}} = - \frac{3 d x}{2 x}$
$- \frac{1}{v} = - \frac{3}{2} ln ∣ x ∣ + C$
$- \frac{x}{y} = \frac{- 3}{2} ln ∣ x ∣ + C$
$x = e, y = \frac{e}{3}$
$C = - \frac{3}{2}$
When $x = 1, y = \frac{2}{3}$

Q. If y=y(x) is the solution of the differential equation 2x2dxdy​−2xy+3y2=0 such that y(e)=3e​, then y(1) is equal to

31​

32​

23​

3

dxdy​−xy​=−23​(xy​)2 y=vx v2dv​=−2x3dx​ −v1​=−23​ln∣x∣+C −yx​=2−3​ln∣x∣+C x=e,y=3e​ C=−23​ When x=1,y=32​

Q. If $y = y (x)$ is the solution of the differential equation $2 x^{2} \frac{d y}{d x} - 2 x y + 3 y^{2} = 0$ such that $y (e) = \frac{e}{3}$ , then $y (1)$ is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{3}{2}$

$3$

$\frac{d y}{d x} - \frac{y}{x} = - \frac{3}{2} (\frac{y}{x})^{2}$
$y = vx$
$\frac{d v}{v ^{2}} = - \frac{3 d x}{2 x}$
$- \frac{1}{v} = - \frac{3}{2} ln ∣ x ∣ + C$
$- \frac{x}{y} = \frac{- 3}{2} ln ∣ x ∣ + C$
$x = e, y = \frac{e}{3}$
$C = - \frac{3}{2}$
When $x = 1, y = \frac{2}{3}$