Ordinary differential equation

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Ordinary differential equation

Definition

Any equation, which involves $x, \, y$ and one or more derivatives (the highest order being $n$), is an ordinary differential equation of the $n$-th order. Example:
$$\begin{align} \frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) &= 12x^2 - 2 \\ \frac{dy}{dx} = \int (12x^2 - 2) \, dx &= 4x^2 - 2x + A \\ \color{red}{y} = \int (4x^2 - 2x + A) \, dx &= \color{red}{x^4 - x^2 + Ax + B} \end{align}$$
The highlighted part is the general solution. Putting in different $A$ and $B$ will give a particular solution. A general solution of an $n$-th order ODE will have $n$ parameters.

First order ODE

Trivial, separable, homogeneous and linear:
$$\begin{array}{ll} \color{red}{y' = f(x)} & y = \int f(x) \, dx \\ \color{red}{y' = f(x) \, g(y)} & \int g(y)^{-1} \, dy = \int f(x) \, dx \\[8pt] \color{red}{y' = f\left(\frac{y}{x}\right)} & \left\{ \begin{array}{l} y = vx \\ v + \frac{dv}{dx} x = f(v) \end{array} \right. \\[8pt] \color{red}{y' + P(x)y = Q(x)} & \left\{ \begin{array}{l} \mu(x) = e^{\int P(x) \, dx} \\ \mu(x) \, y = \int \mu(x) \, Q(x) \, dx \end{array} \right. \end{array}$$

Second order ODE

$y$-less:
$$\begin{array}{ll} \color{red}{ay'' + by' = f(x)} & \left\{ \begin{array}{l} u = \frac{dy}{dx} \\ a\frac{du}{dx} + bu = f(x) \end{array} \right. \end{array}$$
Homogeneous linear:
$$\begin{array}{ll} \color{red}{ay'' + by' + cy = 0} & am^2 + bm + c = 0 \quad \left\{ \begin{array}{ll} \text{2 roots } \alpha \text{ and } \beta & y = Ae^{\alpha x} + Be^{\beta x} \\ \text{1 root } \alpha & y = Ae^{\alpha x} + Bxe^{\alpha x} \\ \text{2 roots } \alpha \pm i\beta & y = e^{\alpha x}(A\cos(\beta x) + B\sin(\beta x)) \end{array} \right. \end{array}$$
Inhomogeneous linear, where $y_{CF}$ is a solution for $f(x) = 0$ (homogeneous), $\, y_{PI}$ can not contain terms of the same type as $y_{CF}$ (multiply by $x$ or $x^2$ in case it does) and $y_{PI}$ has to be solved so it consists of resolved constants only:
$$\begin{array}{ll} \color{red}{ay'' + by' + cy = f(x)} & f(x) = \quad \left\{ \begin{array}{ll} a_n x^n + \cdots + a_1 x + a_0 & y_{PI} = C_n x^n + \cdots + C_1 x + C_0 \\ k \, e^{nx} & y_{PI} = C e^{nx} \\ q \cos nx + r \sin nx & y_{PI} = C \cos nx + D \sin nx \end{array} \right. \\[8pt] & y_{PI}'' + y_{PI}' = f(x) \\ & y = y_{PI} + y_{CF} \end{array}$$