Solve the differential equation 2xy - y2 - 2x2 dydx - 0-

dydx y^22x^2=yx Let y=vx then dydx=v+xdvdx hence v+xdvdx v^22=v xdvdx=v^22 Hence v^ 2dv=0.5x^ 1dx Integrating both sides give us ...

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Standard XII

Mathematics

Particular Solution of a Differential Equation

Solve the dif...

Question

Solve the differential equation $2 x y + y^{2} - 2 x^{2} \frac{d y}{d x} = 0$ .

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Solution

$\frac{d y}{d x} - \frac{y^{2}}{2 x^{2}} = \frac{y}{x}$
Let $y = v x$ then $\frac{d y}{d x} = v + \frac{x d v}{d x}$
hence
$v + \frac{x d v}{d x} - \frac{v^{2}}{2} = v$
$\frac{x d v}{d x} = \frac{v^{2}}{2}$
Hence
$v^{- 2} d v = 0.5 x^{- 1} d x$
Integrating both sides give us
$- v^{- 1} = 0.5 ln x + c$
Or
$\frac{- x}{y} = 0.5 ln x + c$

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Solve the differential equation 2xy+y2−2x2dydx=0.

dydx−y22x2=yxLet y=vx then dydx=v+xdvdxhence v+xdvdx−v22=vxdvdx=v22Hence v−2dv=0.5x−1dxIntegrating both sides give us −v−1=0.5lnx+cOr −xy=0.5lnx+c

Solve the differential equation $2 x y + y^{2} - 2 x^{2} \frac{d y}{d x} = 0$ .