Question

Найти общее решение уравнений
y'-y*cosX=(√X)*e^sinX

Answer 1

$y'-y*cosx=sqrt{x}*e^{sinx}\y=uv;y'=u'v+v'u\u'v+v'u-uvcosx=sqrt{x}*e^{sinx}\u'v+u(v'-vcosx)=sqrt{x}*e^{sinx}\begin{cases}v'-vcosx=0\u'v=sqrt{x}*e^{sinx} end{cases}\frac{dv}{dx}-vcosx=0|*frac{dx}{v}\frac{dv}{v}-cosxdx=0\frac{dv}{v}=cosxdx\intfrac{dv}{v}=int cosxdx\ln|v|=sinx\v=e^{sinx}\frac{du}{dx}e^{sinx}=sqrt{x}*e^{sinx}\frac{du}{dx}=sqrt{x}|*dx\du=sqrt{x}dx\int du=int sqrt{x}dx\u=frac{2sqrt{x^3}}{3}+C\y=(frac{2sqrt{x^3}}{3}+C)*e^{sinx}$