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Solve the given differential equation by using an appropriate substitution. the de is homogeneous. (y2 yx) dx − x2 dy = 0

Sagot :

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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