Question

solve the Diffretial equation, y-x(dy/dx) =x+y(dy/dx) .​

Answer 1

Answer:

Re-arranging the differential equation gives us

dx

dy

=

x

y

+

x

y

2

+x

2

dx

dy

=

x

y

+

x

2

y

2

+x

2

Let y=vx, then

dx

dy

=v+

dx

xdv

Hence

v+

dx

xdv

=v+

1+v

2

dx

xdv

=

1+v

2

∫

1+v

2

dv

=∫

x

dx

ln(v+

1+v

2

)=lnx+C

Or

ln(y+

x

2

+y

2

)−lnx=lnx+c

ln(y+

x

2

+y

2

)=2lnx+c