(sqrt(xy)-sqrt(x))dx+(sqrt(xy)+sqrt(y))dy=0 решить. Created by Алекс102. matematika-ru

(sqrt(xy)-sqrt(x))dx+(sqrt(xy)+sqrt(y))dy=0 решить...

Алекс102 @Алекс102

August 2022 1 11 Report

(sqrt(xy)-sqrt(x))dx+(sqrt(xy)+sqrt(y))dy=0 решить

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yugolovin

Verified answer

$\sqrt{x}( \sqrt{y}-1)\, dx + \sqrt{y}( \sqrt{x} +1)\, dy=0;$

y=1 - решение;

$\frac{ \sqrt{x} }{ \sqrt{x} +1}\, dx +\frac{ \sqrt{y} }{ \sqrt{y} -1}\, dy=0;$

$\sqrt{x} +1=p; \sqrt{y} -1=q; x=(p-1)^2; dx=2(p-1)dp; y=(q+1)^2; dy=2(q+1)dq;$

$\frac{p-1}{p}2(p-1)\, dp +\frac{q+1}{q}2(q+1)\, dq=0$ ;

$\int (q+2+\frac{1}{q})\, dq+\int(p-2+\frac{1}{p})\, dp=0; \frac{q^2}{2}+2q+\ln|q|+\frac{p^2}{2}-2p+\ln |p|=C;$

$\frac{(\sqrt{y}-1)^2}{2}+2(\sqrt{y}-1)+\ln|\sqrt{y}-1|+\frac{(\sqrt{x}+1)^2}{2}+
2(\sqrt{x}+1)+$
$\ln(\sqrt{x}+1)=C$

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