## First Order Linear ODE ### Homogenous $ \dot{y} + p(t) y = 0 $ Solve with separation of variables ### Inhomogeneous $ \dot{y} + p(t) y = q(t) $ #### Variation of Parameters 1. Find a nonzero solution, $y_{h}$ to the associated homogenous equation $\dot{y}_{h}+p(t)y_{h}(t) = 0$ 2. Solve $\dot{u}=\frac{q}{y_{h}}$ for $u(t)$ by integration 3. The general solution is $y=u(t)y_{h}(t)$ ## Second Order Linear ODE with constant coefficients ### Homogenous $ m\ddot{y} + b\dot{y} + ky = 0 $ where $m$, $b$, and $k$ are real constants #### Superposition All linear combinations of homogenous solutions are homogenous solutions Does not hold for nonlinear diff eq’s #### General Solution All solutions are expressed by the following, where $y_{1}(t)$ and $y_{2}(t)$ are solutions to the ODE, and $c_{1}$ and $c_{2}$ take any constant values $ y(t) = c_{1}y_{1}(t) + c_{2}y_{2}(t) $ #### Characteristic Polynomial Assuming that the solution takes the form of $e^{rt}$, the following polynomial must equal zero: $ p(r) = mr^{2} + br + k $ #### Two Real Roots $e^{r_{1}t}$ and $e^{r_{2}t}$ are solutions, giving the general solutions $ y(t) = c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t} $ Use initial conditions $y(0)$ and $\dot{y}(0)$ to solve for $c_{1}$ and $c_{2}$ #### Complex Roots If the roots of the characteristic polynomial are $a+bi$ and $a-bi$, $ \mathrm{Re}(e^{(a+bi)t}) =e^{at}\cos (bt) $ $ \mathrm{Im}{(e^{(a+bi)t})} = e^{at}\sin (bt) $ are real solutions, giving the general solution $ x(t) = c_{1}e^{at}\cos(bt) + c_{2}e^{at}\sin(bt) $ ## Higher Order Linear ODEs with Constant Coefficients ### Homogenous linear ODE $ a_n y^{(n)} + \cdots + a_1 \dot{y} + a_0 y = 0$ where $a_n, \ldots, a_0$ are real constants 1. Write down the **characteristic equation** $ a_n r^n + \cdots + a_1 r + a_0 = 0, $ 2. Factor $P(r)$ as $ P(r) = a_n (r - r_1)(r - r_2) \cdots (r - r_n) $ 3. If $r_1, \ldots, r_n$ are **distinct**, the general solution is $ c_1 e^{r_1 t} + \cdots + c_n e^{r_n t}. $ 4. If $r_1, \ldots, r_n$ are **not distinct**, then replace $ e^{r t}, e^{r t}, e^{r t}, \ldots, e^{r t} $ by $ e^{r t}, te^{r t}, t^2 e^{r t}, \ldots, t^{m-1} e^{r t}, $ where $m$ is the number of times the root $r$ is repeated. ### Inhomogeneous Linear ODE $ p_n(t) y^{(n)} + \ldots + p_0(t) y = q(t) $ 1. Find the general solution $y_{h}$ of the associated homogeneous equation: $ p_n(t) y_{h}^{(n)} + \ldots + p_0(t) y_{h} = 0 $ 2. Find any particular solution $y_{p}$ to the inhomogeneous ODE 3. Use superposition to obtain the general solution to the inhomogeneous ODE: $ y=y_{p}+y_{h} $ #### Exponential Response Formula (ERF) For any polynomial $P$ and any number $r$ such that $P(r)\neq0$, $ \frac{1}{P(r)}e^{rt} ~\text{ is a particular solution to }~ P(D)y = e^{rt} $ #### Generalized ERF If $P$ is a polynomial and $r_{0}$ is a number such that $ P(r_{0})=P'(r_{0})=\dots=P^{(m-1)}(r_{0})=0 $ and $P^{(m)}(r_{0})\neq0$, then $ y_{p} = \frac{1}{P^{(m)}(r_{0})}t^{m}e^{r_{0}t} ~\text{ is a particular solution to }~ P(D)y = e^{r_{0}t} $ #### Complex Replacement Method for finding a particular solution to an inhomogeneous linear ODE $ P(D)y = \cos\omega t, $ where $P$ is a real polynomial, and $\omega$ is a real number. 1. Write $\cos(\omega t)$ as $\text{Re}(e^{j\omega t})$ $ P(D)y = \mathrm{Re}(e^{i\omega t}) $ 2. Replace the right-hand side of the differential equation with the complex exponential $e^{j\omega t}$ $ P(D)y = e^{i\omega t} $ 3. Use ERF (or generalized ERF) to find a particular solution $z_p$ to the complexified ODE: $ z_p = \frac{e^{i\omega t}}{P(i\omega)} $ 4. Compute $y_p = \text{Re}(z_p)$. Then, $y_p$ is a particular solution to the original ODE. ## Definitions ### Span $ \text{Span}(f_{1},\dots, f_{n})=\{\text{all functions }c_{1}f_{1}+\dots +c_{n}f_{n}\text{ where } c_{1},\dots,c_{n} \text{ are real numbers}\} $ ### Vector Space Then $S$ is a **real vector space** if all of the following are true: 1. There is a **zero vector** in $S$ There is a vector $\mathbf{0}$ in $S$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for every vector $\mathbf{v}$ in $S$. 2. **Closed under scalar multiplication** Multiplying any one vector in $S$ by a real number gives another vector in $S$ 3. **Closed under vector addition** Adding any two vectors in $S$ gives another vector in $S$ ### Linear Independence Vectors are **linearly** **dependent** (think redundant) if at least one of them is a linear combination of the others. Otherwise, call them **linearly** **independent**. ### Dimension The **dimension** of a vector space is the number of vectors in **any** basis. The dimension of the space of solutions to an $n^\text{th}$ order homogeneous ODE with constant coefficients is $n$. ### Basis **A basis** of a vector space $S$ is a list of vectors $v_1, v_2, …, v_n$, such that 1. $\text{Span}(v_1, v_2, …, v_n) = S$, and 2. The vectors $v_1, v_2, …, v_n$ are **linearly independent**.