Ordinary Differential Equation

$y'=F(x,y)$, $F(x, y)=0$ are equilibrium solutions.

Separable ODE

$\frac{dy}{dx} = F(x, y) = \frac{P(x)}{Q(y)}$

$\int Q(y)dy = \int P(x)dx + C$

First Order Linear ODE

$y' + p(x)y = q(x)$

The Method of Integrating Factors

$\mu(x) = e^{\int p(x)dx}$

$y(x) = \frac{1}{\mu(x)} [\int \mu(x)q(x)dx + C]$

Modeling

$y' = ay + b$ $y = (y_0 + \frac{b}{a})e^{ax} - \frac{b}{a}$

$y' = ky$ $y = y_0e^{kt}$

Logistic Equation

$y' = ry(1-\frac{y}{L})$

$y = \frac{LA}{A+e^{-rt}}, A = \frac{y_0}{L-y_0}$

$y = 0$ is unstable $y = L$ is asymptotically stable

Autonomous ODE

$y'=F(y)$ $y(t) = a, F(a)=0$ are equilibrium solutions.

Phase Diagram: The sign of $F(y)$ indicates increasing or decreasing.

Classification of Equilibria:

1. $F'(a) < 0$ asymptotically stable
2. $F'(a) > 0$ unstable
3. $F'(a) = 0$ semi-stable or no conclusion

Existence and Uniqueness Theorem for Linear ODE

$y' + p(t)y = q(t), y(t_0) = y_0$

$p(t)$ and $q(t)$ are continuous functions and $t_0$ is on that interval.

$y' = F(t, y), y(t_0) = y_0$

$F, \frac{dF}{dy}$ are continuous.

Euler's Method $y = y_0 + F(x_0, y_0)(x - x_0)$

Second Order Linear ODE

$y'' + p(t)y' + q(t)y = g(t)$ $p, q, g$ are continuous functions.

Constant Coefficients

Homogeneous

$ay'' + by' + cy = 0$

Superposition Principle: $y = c_1y_1 + c_2y_2$, $y = y_h + y_p$ Wronskian: $W(y_1, y_2) = \begin{vmatrix} y_1(t_0) & y_2(t_0) \\ y_1'(t_0) & y_2'(t_0) \end{vmatrix}$ Characteristic Equation: $ar^2 + br + c = 0$

1. $y = [e^{r_1t},\ e^{r_2t}]$
2. $y = [e^{rt},\ te^{rt}]$
3. $r = \lambda \pm i\mu$, then $y = [e^{\lambda t}\cos\mu t,\ e^{\lambda t}\sin\mu t]$

Euler's Formula: $e^{i\theta} = \cos\theta + i\sin\theta$, $e^{i\pi} + 1 = 0$

Amplitude and Phase Angle:

$$c_1\cos(\omega t) + c_2\sin(\omega t) = A\cos(\omega t - \theta) \\ A = \sqrt{c_1^2 + c_2^2},\ \cos\theta = \frac{c_1}{\sqrt{c_1^2 + c_2^2}}$$

Inhomogeneous

Undetermined Coefficients

$g(t)$ is

1. polynomial, $y_p = A_0+A_1t+\dots+A_nt^n$
2. exponential, $y_p = Ae^{\alpha t}$
3. trigonometric, $y_p = A\sin\beta t + B\cos\beta t$

When exception happens, multiply by $t$

Variation of Parameters

Assumption: $v_1'y_1 + v_2'y_2 = 0$

$y_p = v_1y_1 + v_2y_2$ $W = y_1y_2' - y_2y_1' \\ v_1 = -\int\frac{y_2g}{W}dt \\ v_2 = \int\frac{y_1g}{W}dt \\$

Modelling

Spring Mass System: $mx'' + rx' + kx = F(t)$ RLC Circuit: $LQ'' + RQ' + \frac{1}{C}Q = E(t)$

High Order Linear ODE with constant coefficients are the same with second order.

First Order Systems of ODEs

$x' = Ax + g$

1. Matrix $A$'s eigenvalues and eigenvectors are $r_1, v_1$ and $r_2, v_2$ $x = [e^{r_1t}\vec{v_1},\ e^{r_2t}\vec{v_2}]$

2. $r, v, u$ $(A-rI)u = v$ $x = [e^{rt}v, e^{rt}(tv + u)]$

3. $r_{1,2} = \lambda\pm i\mu$ and $\vec{v}_{1,2} = \vec{a}\pm i\vec{b}$, then $x_1 = e^{\lambda t}[\cos(\mu t)\vec{a} - \sin(\mu t)\vec{b}] \\ x_2 = e^{\lambda t}[\sin(\mu t)\vec{a} + \cos(\mu t)\vec{b}]$

Phase Portrait: test $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $x_2' > 0$: counterclockwise; $x_2' < 0$: clockwise test the first column or $A$, $A_{2,1} > 0$: counterclockwise; $A_{2,1} < 0$: clockwise

Stability of equilibria

• Real eigenvalues
  • $r_1, r_2 < 0$, asymptotically stable node
  • $r_1, r_2 > 0$, unstable node
  • $r_1r_2 < 0$, unstable saddle
• Complex eigenvalues
  • $\lambda = 0$, stable center
  • $\lambda < 0$, spiral sink
  • $\lambda > 0$, spiral source

High order linear DEs can be transformed to first order linear system.

Nonhomogenous

If $g$ is a constant vector, a particular equilibrium solution is $x_p = -A^{-1}g$

Undetermined Coefficients

Variation of Parameters

The fundamental matrix of homogeneous is $\Phi(t)$ $\mu = \int \Phi^{-1}g dt$ $x_p = \Phi \mu$

Autonomous

$\vec{x'} = A\vec{x} + \vec{g} = 0$

The stability can be determined by $A$

Nonlinear Autonomous System of DEs

$x' = f(x, y) \\ y' = g(x, y) \\$ $A = \begin{bmatrix} f_x(x_0, y_0) & f_y(x_0, y_0) \\ g_x(x_0, y_0) & g_y(x_0, y_0) \\ \end{bmatrix}$

Boundary Value Problem

$y(a) = y_0, y(b) = y_1$

nonhomogeneous has unique solution ↔ homogeneous has only trivial solution nonhomogeneous has no/infinite solution ↔ homogeneous has nontrivial solution