The general solution of y2 d x-x2-x y-y2 d y-0 is -EAMCET 2003-A- sinh -1x-y-log y-c-0B- logy-x2-y2-log y-c-0C- tan -1x-y-log y-c-0D- 2 tan -1x-y-log x-c-0

The correct option is C dx/dy + x^2 xy + y^2/y^2 = 0 dx/dy + ( x/y )^2 ( x/y ) + 1 = 0 Put v = x/y ⇒ x = vy ⇒dx/dy = v + y dv/dy v + y dv/dy + v^2 v + ...

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Byju's Answer

Standard XII

Mathematics

Solving Homogeneous Differential Equations

The general s...

Question

The general solution of $y^{2} d x + (x^{2} - x y + y^{2}) d y = 0$ is

[EAMCET 2003]

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Solution

The correct option is A

$\frac{d x}{d y} + \frac{x^{2} - x y + y^{2}}{y^{2}} = 0 \frac{d x}{d y} + {(\frac{x}{y})}^{2} - (\frac{x}{y}) + 1 = 0$
Put $v = x / y \Rightarrow x = v y \Rightarrow \frac{d x}{d y} = v + y \frac{d v}{d y} v + y \frac{d v}{d y} + v^{2} - v + 1 = 0 \Rightarrow \frac{d v}{v^{2} + 1} + \frac{d y}{y} = 0 \Rightarrow \int \frac{d v}{v^{2} + 1} + \int \frac{d y}{y} = 0 \Rightarrow t a n^{- 1} (v) + l o g y + C = 0 \Rightarrow t a n^{- 1} (x / y) + l o g y + c = 0.$

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Similar questions

Q. The general solution of

y^{2} d x + (x^{2} - x y + y^{2}) d y = 0

Q. The solution of

(y + x + 5) d y = (y - x + 1) d x

Q. The solution of the differential equation

\frac{d y}{d x} = \frac{x^{2} + x y + y^{2}}{x^{2}}

, is
(a)

\tan^{- 1} (\frac{x}{y}) = \log y + C

(b)

\tan^{- 1} (\frac{y}{x}) = \log x + C

(c)

\tan^{- 1} (\frac{x}{y}) = \log x + C

(d)

\tan^{- 1} (\frac{y}{x}) = \log y + C

Q. If

{log}_{y} x + {log}_{x} y = 7

then

({log}_{y} x)^{2} + ({log}_{x} y)^{2}

is equal to

I

f (x, y) = \frac{1}{x^{2}} + \frac{1}{x y} + \frac{log x - log y}{x^{2} + y^{2}}

, then

x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}

is equal to

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The general solution of y2 dx+(x2−xy+y2)dy=0 is [EAMCET 2003]

The correct option is A dxdy+x2−xy+y2y2=0dxdy+(xy)2−(xy)+1=0 Put v=x/y⇒x=vy⇒dxdy=v+ydvdyv+ydvdy+v2−v+1=0⇒dvv2+1+dyy=0⇒∫dvv2+1+∫dyy=0⇒tan−1(v)+log y+C=0⇒tan−1(x/y)+log y+c=0.

The general solution of $y^{2} d x + (x^{2} - x y + y^{2}) d y = 0$ is

[EAMCET 2003]