tex]...

olgas2000 @olgas2000

August 2022 1 6 Report

Помогите, пожалуйста решить!!!!
Найдите общее решение дифференциального уравнения первого порядка.
[tex](x-y)y-x^{2} y'=0[/tex]

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NNNLLL54

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$(x-y)y-x^2y'=0\\\\y'= \frac{(x-y)y}{x^2}\; ,\; \; \; y'= \frac{xy-y^2}{x^2}\; ,\; \; \; y'= \frac{y}{x}- (\frac{y}{x})^2\\\\u= \frac{y}{x}\; ,\; \; y=ux\; ,\; \; y'=u'x+ux'=u'x+u\; \;
 \; (x'=1)\\\\u'x+u=u-u^2\\\\u'= -\frac{u^2}{x} \; ,\quad \frac{du}{dx} =-\frac{u^2}{x} \; ,\quad \int \frac{du}{u^2}=-\int \frac{dx}{x} \\\\-\frac{1}{u}=-\ln|x|+C\\\\- \frac{x}{y} =-ln|x|+C\\\\y=\frac{x}{ln|x|-C}$

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