Question

Линейное дифференциальное уравнение

Answer 1

а)

$y' + 3e^x = 0\\y' = -3e^x\\dy = -3e^x dx\\\int {}dy = -3\int{e^x}\,dx\\y = -3e^x + c_1$

б)

$ydx = (1 + x^2)dy\\\frac{dx}{1 + x^2} = \frac{dy}{y}\\\int {\frac{dx}{1 + x^2}} = \int {\frac{dy}{y}}\\\arctan{x} + c_1= \ln y\\y = c_2e^{\arctan{x}}$

в)

$dy - xy^2\sqrt{2x^2 + 3}dx = 0\\\frac{dy}{y^2} = \frac{xdx}{\sqrt{2x^2 + 3}}\\\int\frac{dy}{y^2} = \int\frac{xdx}{\sqrt{2x^2 + 3}}\\\int\frac{dy}{y^2} = \frac14\int\frac{d(2x^2 + 3)}{\sqrt{2x^2+3}}\\-\frac{1}{y} = \frac12 \sqrt{2x^2 + 3} + c_1\\y = \frac{-1}{\frac12 \sqrt{2x^2 + 3} + c_1}$