Question

Solve the differential equation (X²+Y²)dy/dx=xy

Answer 1

Answer:

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dx

dy

=

x

2

+y

2

xy

Put y=vx...(1)∴

dx

dy

=v+x

dx

dv

...(2)

∴v+x

dx

dv

=

x

2

+v

2

x

2

x.vx

=

1+v

2

v

...(3)

x

dx

dv

=

1+v

2

v

−v,...(4)

x

dx

dv

=

1+v

2

v−v−v

3

=

1+v

2

−v

3

∴(

−v

3

1+v

2

)dv=

x

dx

or (

v

3

−1

−

v

1

)dv=

x

dx

Integrate both sides,

2v

2

1

−logv=logx+c,

or

2v

2

1

=logvxk, where c=logk

or

2

1

y

2

x

2

=log

x

y

x.k=logky.∴e

x

2

/2y

2

=ky

Step-by-step explanation:

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