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The curve amongst the family of curves responded by the differential equation, $(x^2 - y^2)dx +2xy\: \: \: dy = 0$ which passes through (1,1), is:
Option 1)
A hyperbola with the transverse axis along the x-axis
Option 2)
A circle with centre on the x-axis
Option 3)
an ellipse with the major axis along the y-axis.
Option 4)
A circle with centre on the y-axis

Answers (1)

best_answer

Homogeneous Differential Equation -

A function

$f\left ( x,y \right )$

is called homogeneous function of degree n, if

$f\left (dx,dy \right )=d^{n}\left ( x,y \right )$

- wherein

eg:

$f\left ( x,y \right )=x^{2}y^{2}-xy^{3}$

Homogeneous Differential Equation -

Put

$\frac{y}{x}=v$

$\frac{dy}{dx}=v+\frac{xdv}{dx}$

-

$(x^{2}-y^{2})dx+2xy\: dy=0$

The D.E. can be written as

$\frac{dy}{dx}=\frac{y^{2}-x^{2}}{2xy}$

from the concept

put $\frac{y}{x}=v$

$=>\frac{dy}{dx}=v+x\frac{dv}{dx}$

$=>\int \frac{2v}{v^{2}+1}dv=\int \frac{-dx}{x}$

On Integrating

$\ln (v^{2}+1)=-\ln (x)+C$

$(y^{2}+x^{2})=Cx$

Passes through (1,1)

$=>C=2$

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

Option 1)

A hyperbola with the transverse axis along the x-axis

Option 2)
A circle with centre on the x-axis
Option 3)

an ellipse with the major axis along the y-axis.

Option 4)

A circle with centre on the y-axis

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