The solution for the given differential equation is lnlxl = [tex]\frac{-x}{y} +C[/tex]
Given,
[tex](y^{2} +y^{x} )dx - x^{2} dy =0[/tex]
Here,
x = vy, dx = vdy + ydv
y = ux, dy = udx + xdu
Then,
[tex]((ux)^{2} +(ux)x)dx-x^{2} (udx+xdu)=0\\=u^{2} x^{2} dx+ux^{2} dx-x^{2} udx-x^{3} du=0\\=u^{2} x^{2} dx=x^{3} du\\=\frac{x^{2} }{x^{3} } dx=\frac{1}{u^{2} } du[/tex]
Now,
[tex]\int\limits^a_b {\frac{1}{x} } \, dx =\int\limits^a_b {u^{-2} } \, du[/tex]
l[tex]n[/tex]l[tex]x[/tex]l[tex]=\frac{u^{-1} }{(-1)} +C[/tex]
l[tex]n[/tex]l[tex]x[/tex]l[tex]=\frac{-1}{u} +C[/tex]
y = ux
u = [tex]\frac{y}{x}[/tex]
That is l[tex]n[/tex]l[tex]x[/tex]l [tex]=\frac{-x}{y} +C[/tex]
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