Question

Решите дифференциальные уравнения бернулли y'+x^2*y=x^2

Answer 1

$y`+x^2*y=x^2\\y`+x^2*y-x^2=0\\\frac{dy}{dx}+x^2(y-1)\\\frac{dy}{dx}=x^2(1-y)|*\frac{dx}{(1-y)}\\\frac{dy}{1-y}=x^2dx\\\int\frac{dy}{1-y}=\int x^2dx\\-\int\frac{d(1-y)}{1-y}=\int x^2dx\\-ln(1-y)=\frac{x^3}{3}+c\\ln(1-y)=-\frac{x^3}{3}-c\\1-y=e^{-\frac{x^3}{3}-c}\\y=1-ce^{-\frac{x^3}{3}}$