Question

решите пожалуйста диф.уравнения

Answer 1

$1)\; xy'=y-xe^{y/x}\\\\y'=\frac{y}{x}-e^{y/x}\\\\t=y/x\; ,\; y=tx,\; y'=t'x+t\\\\t'x+t=t-e^{t}\\\\\frac{dt}{dx}=-\frac{e^{t}}{x}\\\\\int \frac{dt}{e^{t}}=-\int \frac{dx}{x}\\\\-e^{-t}=-ln|x|+C\\\\-e^{-\frac{y}{x}}=-ln|x|+C$

$y(1)=2,\; -e^{-\frac{2}{1}}=-ln1+C,\; \to \; C=-\frac{1}{e^2}\\\\e^{-\frac{y}{x}}=ln|x|+\frac{1}{e^2}$

$2)\; y'+y\cdot tgx=\frac{1}{cosx}\\\\y=uv\\\\u'v+uv'+uvtgx=\frac{1}{cosx}\\\\u'v+u(v'+vtgx)=\frac{1}{cosx}\\\\a)\; v'+vtgx=0\\\\\int \frac{dv}{v}=-\int tgx\, dx\\\\ln|v|=ln|cosx|\\\\v=cosx\\\\b)\; u'cosx=\frac{1}{cosx}$

$\int du=\int \frac{dx}{cos^2x}\\\\u=tgx+C\\\\c)\; y=cosx(tgx+C)$