Question

Given the Homogeneous equation x^2ydy+xy^2dx=0, use y=ux, u=y/x and dy=udx+xdu to solve the differential equation. Solve for y.

Answer 1

`y=(C)/(x)`.

Explanation:

Given homogeneous equation

`x^2ydy+xy^2dx=0`

`\frac{\mathrm{d}y}{\mathrm{d}x}=-(xy^2)/(x^2y)`

Substitute y=ux ,
`u=(y)/(x)`

`\frac{\mathrm{d}y}{\mathrm{d}x}=-(y)/(x)`

Now,

`u+x\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}x}`

`u+x\frac{\mathrm{d}u}{\mathrm{d}x}=-u`

`\frac{\mathrm{d}u}{\mathrm{d}x}=-2u`

`(du)/(u)=-(dx)/(x)`

Integrating both side we get

lnu=-2lnx+lnC

Where lnC= integration constant

`lnu+ln{x}^2=lnC`

`lnux^2=lnC`

Cancel ln on both side

`ux^2=C`

Substitute
`u=(y)/(x)`

Then we get

xy=C

`y=(C)/(x)`.

Answer:
`y=(C)/(x)`.