Types of ODE

linear: only linear y terms
homogeneous: $y' + p(x)y = 0$ no term without y
autonomous: no x in equation

Existence and uniqueness:

If $f \in C(D)$ and $(x_0, y_0) \in D$ then at least ONE solution.
If $\frac{\delta f(y)}{\delta y} \in C(D)$ $\implies$ unique solution.

Linear 1’st order ODE

form: $y'(x) + p(x)y(x) = f(x)$
$f(x) = 0$ $\implies$ homogeneous ODE $\implies$ solve by speration of variables
$f(x) \neq 0$ $\implies$ inhomogeneous general solution: $e^{-P(x)} \cdot (\int e^{P(x)} \cdot f(x) dx + C)$

get general solution using superposition:
$y = y_{homo} + y_{p}$
ansatz: $y_p(x) = c(x) \cdot y_{homo}$ Insert ansatz into the inhomogeneous ODE $\implies$
$c(x) = \int e^{P(x)} \cdot f(x) dx +C$

Bernoulli DE(non linear)

form: $y'(x) +p(x)y(x) = f(x) \cdot y^{\alpha}$ ansatz for $\alpha \neq {1;0}$: $u(x) = y^{1- \alpha}(x) \implies$solve this homogeneous ODE

Order reduction

form: Second order autonomous ODE
ansatz: $v(y) = \frac{dy}{dx} = y' \implies y'' = \frac{dy'}{dx} = v' \cdot v$^

Second-order linear DE

form: $y''(x) + p(x)y'(x) + q(x)y(x) = r(x)$