Question

Решить дифференциальное уравнение y'=2y/x+1 + (e^x) * (x+1)^2

Answer 1

$y'= \frac{2y}{x+1}+e^{x}(x+1)^2\\\\y'-\frac{2y}{x+1}=e^{x}(x+1)^2\; \; ,\quad y=uv\; ,\; y'=u'v+uv'\\\\ u'v+uv'- \frac{2uv}{x+1}=e^{x}(x+1)^2\\\\u'v+u(v'- \frac{2v}{x+1})=e^{x}(x+1)^2\\\\1)\; \; \frac{dv}{dx}-\frac{2v}{x+1}=0\; ,\; \; \int \frac{dv}{v}=2\int \frac{dx}{x+1}\; ,\; ln|v|=2ln|x+1|\\\\v=(x+1)^2\\\\2)\; \; u'v=e^{x}(x+1)^2\\\\ \frac{du}{dx}\cdot (x+1)^2=e^{x}(x+1)^2\; ,\; \; \int du=\int e^{x}\, dx\\\\u=e^{x}+C\\\\3)\; \; y=uv\\\\y=(x+1)^2\cdot (e^{x}+C)$