2 Solve the exact differential equation a y2x2dy 2xyxdx 0S

2. Solve the exact differential equation:

a) (y^2+x^2)dy + (2xy+x)dx = 0

ans)

Geometry also helps. 2xy2xy looks like a diagonal term, rotation by 4545 will simplify the equation. Use

x=u+vx=u+v

y=uvy=uv

(substitution of new variables u=(x+y)/2u=(x+y)/2 and v=(xy)/2v=(xy)/2).

We get

2(u2+v2)(du+dv)2(u2v2)(dudv)=02(u2+v2)(du+dv)2(u2v2)(dudv)=0

v2du+u2dv=0v2du+u2dv=0

This is actually as good as it can possibly get. You can separate the variables!

duu2=dvv2duu2=dvv2

1/u+1/v=C1/u+1/v=C

u+v=Cuvu+v=Cuv

Cx=x2y2