ODE collection of first order differential equations

Collection of first order ODE solutions

ODE	Solution	Online-Calculator
$y^{'} (x) + a y (x) = b$	$y (x) = \frac{b}{a} + C e^{- a x}$	y'+ay=b
$y^{'} (x) + a y (x) = c e^{b x}$	$y (x) = \frac{c e^{b x}}{a + b} + k_{1} e^{- a x}$	y'+ay=ce^bx
$y^{'} (x) + 2 x y (x) = x e^{- x^{2}}$	$y (x) = k_{1} e^{- x^{2}} + \frac{1}{2} e^{- x^{2}} x^{2}$ $= e^{- x^{2}} (k_{1} + \frac{x^{2}}{2})$	y'+2xy=xe^(-x^2)
$y^{'} (x) + x y (x) = x$	$y (x) = c_{1} e^{- \frac{x^{2}}{2}} + 1$	y'+xy=x
$y^{'} (x) + y (x) = x$	$y (x) = c_{1} e^{- x} + x - 1$	y'+y=x
$y^{'} (x) = {y (x)}^{2}$	$y (x) = \frac{1}{c_{1} - x}$	y'=y^2
$y^{'} (x) + {y (x)}^{2} = 1$ Separated: $\frac{y^{'} (x)}{1 - {y (x)}^{2}} = 1$ Riccatische Equation: $y^{'} (x) = - {y (x)}^{2} + 1$	$y (x) = \frac{e^{2 x} - e^{2 c_{1}}}{e^{2 c_{1}} + e^{2 x}}$	y'+y^2=1
$y^{'} (x) = (A y (x) - a) (B y (x) - b)$	$y (x) = \frac{b e^{x (a B - A b)} - a C}{B e^{x (a B - A b)} - A C}$	y'=(Ay-a)(By-b)