Answer:
[tex] y=\frac{C}{x}[/tex].
Step-by-step explanation:
Given homogeneous equation
[tex] x^2ydy+xy^2dx=0[/tex]
[tex]\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{xy^2}{x^2y}[/tex]
Substitute y=ux , [tex]u=\frac{y}{x}[/tex]
[tex] \frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{y}{x}[/tex]
Now,
[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]
[tex]u+x\frac{\mathrm{d}u}{\mathrm{d}x}=-u[/tex]
[tex]\frac{\mathrm{d}u}{\mathrm{d}x}=-2u[/tex]
[tex]\frac{du}{u}=-\frac{dx}{x}[/tex]
Integrating both side we get
lnu=-2lnx+lnC
Where lnC= integration constant
[tex]lnu+ln{x}^2=lnC[/tex]
[tex]lnux^2=lnC[/tex]
Cancel ln on both side
[tex]ux^2=C[/tex]
Substitute [tex]u=\frac{y}{x}[/tex]
Then we get
xy=C
[tex]y=\frac{C}{x}[/tex].
Answer:[tex]y=\frac{C}{x}[/tex].
