Question

solve (x2+y2)dx=2xy dy

Answer 1

Answer:

y=±√x2+Cx

Explanation:

Another approach

With equations like these, it is often good to check for exactness

An equation in this form:

(x2+y2)dx−2xydy=0

....can be related to level surface Φ(x,y)=constsuch that dΦ=0=Φxdx+Φydy

So in this case

Φx=x2+y2,Φxy=2y

Φy=−2xy,Φyx=−2y

But because Φxy≠Φyx, the equation is not exact

We can then re-arrange the equation to say:

dydx=x2+y22xy=x2y+y2x

This is a homogeneous equation, ie

dydx=f(x,y)=f(kx,ky) for any constant k

in this case we make a substitution v(x)=yx,y=vx, so that y'=v'x+v and the new equation becomes separable

So
dydx=x2y+y2x

⇒v'x+v=12v+v2

v'x=1−v22v

2v1−v2v'=1x

We integrate wrt x:
∫(2v1−v2v'=1x)dx

⇒∫2v1−v2dv=∫1xdx

Or

−ln∣∣1−v2∣∣=ln|x|+C   [=lnC|x|] where, throughout, C is a generic constant

⇒−1|1−v2|=C|x|

For suitable v and x

1−v2=1Cx

1−(yx)2=1Cx

y2=x2+Cx

y=±√x2+Cx