# Solve the differential equation (x2 + y2)dx- 2xydy = 0 - Mathematics and Statistics

Sum

Solve the differential equation (x2 + y2)dx- 2xydy = 0

#### Solution

(x2 + y2)dx- 2xydy = 0

(x2 + y2) dx = 2xydy

dy/dx = (x^2 + y^2)/(2xy).........(i)

The equation is a homogeneous equation
Let y= vx,
Differentiat ing w.r.t. x, we get,

dy/dx=v+x(dv)/dx

dy/dx=(x^2+y^2)/(2xy) " from "(i)

v+x(dv)/dx=(x^2+(vx)^2)/(2x.(vx))

v+x(dv)/dx=(1+v^2)/(2v)

x(dv)/dx=(1+v^2)/(2v)-v

x(dv)/dx=(1+v^2-2v^2)/(2v)

x(dv)/dx=(1-v^2)/(2v)

(2v)/(1-v^2)dv=1/xdx.......(ii)

Which is in variables separatable form

∴ Integrating both sides, we get

int(2v)/(1-v^2)dv=int1/xdx + c_1

therefore -log|1-v^2|=log|x|+logc

therefore log|x(1-v^2)|=log|c|

therefore x(1-v^2)=c

Resubstituting v=y/x we get

x(1-y^2/x^2)=c

x((x^2-y^2)/x^2)=c

therefore x^2 - y^2 = cx, where c is constant

which is the required general solution

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