Question

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

Answer 1

The solution for the given differential equation is lnlxl = $\frac{-x}{y} +C$

Given,

$(y^{2} +y^{x} )dx - x^{2} dy =0$

Here,

x = vy, dx = vdy + ydv

y = ux, dy = udx + xdu

Then,

$((ux)^{2} +(ux)x)dx-x^{2} (udx+xdu)=0\\=u^{2} x^{2} dx+ux^{2} dx-x^{2} udx-x^{3} du=0\\=u^{2} x^{2} dx=x^{3} du\\=\frac{x^{2} }{x^{3} } dx=\frac{1}{u^{2} } du$

Now,

$\int\limits^a_b {\frac{1}{x} } \, dx =\int\limits^a_b {u^{-2} } \, du$

l$n$l$x$l$=\frac{u^{-1} }{(-1)} +C$

l$n$l$x$l$=\frac{-1}{u} +C$

y = ux

u = $\frac{y}{x}$

That is l$n$l$x$l $=\frac{-x}{y} +C$

Learn more about differential equation here: brainly.com/question/21852102

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