Question

Решить дифференциальное уравнение: y′ − x2y = 2xy

Answer 1

Ответ:

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

$c)\ \ y=e^{\frac{x^3}{3}}\cdot e^{x^2+C}\ \ ,\ \ \ y=e^{\frac{x^3}{3}+x^2+C}\ \ ,\ \ \boxed{\ y=C_1\cdot e^{\frac{x^3}{3}+x^2}\ }\ \ ,\ \ C_1=e^{C}$