Advertisement Remove all ads

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

Sum

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

Advertisement Remove all ads

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

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×