Question

Решите дифференциальное уравнение

Answer 1

$y'+yctgx=\frac{1}{sinx}\\y=uv;y'=u'v+v'u\\u'v+v'u+uvctgx=\frac{1}{sinx}\\u'v+u(v'+vctgx)=\frac{1}{sinx}\\\begin{cases}v'+vctgx=0\\u'v=\frac{1}{sinx}\end{cases}\\\frac{dv}{dx}+vctgx=0|*\frac{dx}{v}\\\frac{dv}{v}=-ctgxdx\\\int\frac{dv}{v}=-\int ctgxdx\\ln|v|=-ln|sinx|\\v=\frac{1}{sinx}\\\frac{du}{sinxdx}=\frac{1}{sinx}\\du=dx\\\int du=\int dx\\u=x+C\\y=\frac{x}{sinx}+\frac{C}{sinx}$
$y(\frac{\pi}{2})=\frac{\pi}{2}\\\frac{\pi}{2}=\frac{\pi}{2}+C\\C=0\\y=\frac{x}{sinx}$