Question

solve 2ydx - xdy = xy^3dy ordinary differential equation ​

Answer 1

Answer:

x

2

ydx−(x

3

+y

3

)dy=0

dy

dx

=

x

2

y

x

3

+y

3

Putting x=vy

dy

dx

=v+y

dy

dv

dy

dx

=

x

2

y

x

3

+y

3

⇒v+y

dy

dv

=

v

2

y

3

v

3

y

3

+y

3

=

v

2

v

3

+1

y

dy

dv

=

v

2

v

3

+1

−v=

v

2

v

3

+1−v

3

=

v

2

1

⇒v

2

dv=

y

dy

(variable separable method)

Integrating both sides

∫v

2

dv=∫

y

dy

⇒

3

v

3

=lny+C

Putting v=

y

x

3y

3

x

3

=lny+C