# Интегралы дз

### С разделяющимися переменными

1)
$$
ydx + (1 + x^2)dy = 0\\
ydx = -(1 + x^2)dy\\
\frac{dx}{1 + x^2} = -\frac{dy}{y}\\
\int\frac{dx}{1 + x^2} = \int-\frac{dy}{y}\\
\frac{ln|1 + x^2|}{2} = -ln||-ln||
$$
2)
$$
y` = \tan{x}\tan{y}\\
\frac{dy}{dx} = \tan{x}\tan{y}\\
\frac{dy}{tan(y)} = dxtan(x)\\
\int\frac{dy}{tan(y)} = \int{dxtan(x)}\\
ln(siny) = -ln(cosx) + C
$$

### Последний слайд

1)
$$
y` = \frac{y}{x} -1\\
\frac{dy}{dx} = \frac{y}{x} -1\\
dy = (\frac{y}{x} -1)dx\\
u = \frac{y}{x};\ y = ux;\ dy = udx + xdu\\
udx + xdu = (u - 1)dx\\
udx + xdu = udx - dx\\
xdu = -dx\\
du = -\frac{dx}{x}\\
\int{du} = \int{-\frac{1}{x}}dx\\
u = = -ln|x| + C\\
\frac{y}{x} = -ln|x| + C\\
y = x(C - ln|x|)
$$

2)

$$
y` = \frac{y}{x} + e^{\frac{y}{x}}\\
dy = dx(\frac{y}{x} + e^{\frac{y}{x}})\\
u = \frac{y}{x};\ y = ux;\ dy = udx + xdu\\
udx + xdu = dx(e^u + u)\\
udx + xdu = e^udx + udx\\
xdu = e^udx\\
\frac{du}{e^u} = \frac{dx}{x}\\
\int\frac{du}{e^u} = \int\frac{dx}{x}\\
\frac{1}{e^u} = ln(x) + C\\
\frac{1}{e^\frac{y}{x}} = ln(x) + C
$$

4)
$$
y` = \frac{x - 2y + 3}{x - 2y + 1}\\
z = x - 2y + 1\\
y = (-z + x + 1)/2. y` = (-z + 1)/2\\
\frac{-z + 1}{2} = \frac{z + 2}{z}\\
а дальше хз
$$