### 1.1: Introduction to Differential Equations **Forms:** $\frac{dy}{dx},\frac{d^{n}y}{dx^{n}}$ or $y',y^{(n)}$ or $\dot{y},\ddot{y}$ or $u_{xy}=\frac{\partial^{2}u}{\partial x\partial y}$. **Definition:** Equation with derivatives of dependent var(s) w.r.t. independent var(s). - **Order:** Highest derivative. **ODE:** 1 independent var. **PDE:** >1 independent var. - **Linear DE:** Linear in dependent var (nonlinear in independent var allowed). **Solution:** $y=\phi(x)$ satisfying DE on an interval. **Implicit solution:** $G(x,y)=0$ (check via implicit diff’n). **Forms:** *General*: $F(x,y,y',\dots,y^{(n)})=0$; *Normal*: $f(x,y,\dots,y^{(n-1)})=y^{(n)}$. ### 1.2: Initial Value Problems **Initial Value Problem (IVP):** DE + initial condition $y(x_0)=y_0$. **Existence/Uniqueness:** If $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continuous in a rectangular region containing $(x_0,y_0)$, then IVP has a **unique solution** in some interval around $x_0$. (No guarantees on solution form, interval size, or uniqueness range, only local uniqueness) ### 2.1: Direction Fields, Autonomous Equations and Critical Points **Unique IVP Solution:** If $y'=f(x,y)$ and $f_y(x,y)$ are continuous on a rectangle, then **unique solution** exists for every $(x_0,y_0)$ in the region. Plotting slopes at grid points yields a **direction field**. **Autonomous DE:** $\frac{dy}{dx}=f(y)$ (no $x$-dependence). - If $f(y)$ is differentiable $\Rightarrow$ $f(y)$ continuous, and $f'(y)=\frac{\partial f}{\partial y}$. - If $f'(y)$ is continuous $\Rightarrow$ satisfies unique IVP criteria. - **Solution method:** Separate variables $\Rightarrow \int \frac{1}{f(y)}dy = \int dx$. **Critical Points:** $y'=0$. If $y=c$ is critical, then $y(x)=c$ is a **constant solution** (appears as horizontal line in direction field). Solutions **cannot cross equilibria**. - **Stable:** Nearby solutions approach from both sides. - **Semi-stable:** Solutions approach from one side only. - **Unstable:** Nearby solutions diverge. ### 2.2: Separable Equations **Separable DE:** $\frac{dy}{dx}=g(x)h(y)$. Solve via separation: $$\int \frac{1}{h(y)}dy = \int g(x)dx.$$ **Exceptional solutions:** Zeroes of $h(y)$, i.e., $y_i$ where $h(y_i)=0$ $\Rightarrow$ constant solutions $y(x)=y_i$ (lost when dividing by $h(y)$). FIND THESE FIRST. **Post-integration:** Constant $C$'s solutions may already include the exceptional solutions (i.e if a $C$ exists such that $y(x)=y_i$). If not, $y_i$ is a **singular solution**: must be included piecewise (e.g., for $y_0=0$). ### 2.3: Linear Equations **Linear First-Order DE:** $a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)$. - **Homogeneous:** $g(x)=0$; **nonhomogeneous:** $g(x)\neq0$. **General *n*-th Order Linear DE:** $a_n(x)\frac{d^n y}{dx^n} + \dots + a_0(x)y = g(x)$. **Solving Homogeneous First-Order:** Separation of variables. **Solving Nonhomogeneous First-Order:** 1. Rewrite in **standard form**: $\frac{dy}{dx} + P(x)y = Q(x)$. 2. Multiply by **integrating factor** $e^{\int P(x)\,dx}$. 3. Left side becomes $\frac{d}{dx}\left(y e^{\int P(x)\,dx}\right)$; right side is $Q(x)e^{\int P(x)\,dx}$. 4. Integrate both sides, solve for $y(x)$. ### 2.7: Linear Models **Newton’s Cooling/Heating:** $\frac{dT}{dt} = -k(T-T_s),\quad T(t_0)=T_0$ $T(t)$ = body temp, $k$ = constant, $T_s$ = surroundings. **Solution:** $T(t)=T_s + (T_0-T_s)e^{-kt}$. *Hint:* General exponential form $y(t)=y(0)e^{kt}$ applies to any $\frac{dQ}{dt}=kQ$ (e.g., population growth). **Logistic Growth (Nonlinear):** $\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$ $K$ = carrying capacity (**stable** critical point). Solve via separation of variables. **Euler’s Method:** Given $\frac{dy}{dx}=f(x,y)$, $y(x_0)=y_0$, approximate via: $y_{n+1} = y_n + h f(x_n,y_n),\quad x_{n+1} = x_n + h$ ### 3.1: Theory of Linear Equations **$n$-th Order Linear DE:** General form: $a_n(x)y^{(n)}+\dots+a_0(x)y=g(x)$. **IVP:** Add $n$ initial conditions $y(x_0)=y_0,\dots,y^{(n-1)}(x_0)=y_{n-1}$. **Existence/Uniqueness:** If $a_0(x),\dots,a_n(x),g(x)$ are continuous on $I=(a,b)$ **and** $a_n(x)\neq0$ $\forall x\in I$, then **unique solution** exists on **entire** $I$ (make coefficient of leading order term $1$ before doing this, interval must contain $x_{0}$). **Warning:** Ensure $a_n(x)\neq0$ on $I$ (e.g., avoid $x=0$ if $a_n(x)=x$, make sure leading term does not vanish if interval has $0$). **Differential Operator:** $D^n y = y^{(n)}$. Define $L = a_n(x)D^n + \dots + a_0(x)$. General non-homogeneous ODE: $L[y] = g(x)$. **Superposition Principle (Homogeneous Case, $L[y]=0$):** If $y_1,\dots,y_k$ solve $L[y]=0$, then **any linear combo** $y = c_1y_1 + \dots + c_ky_k$ also solves it. Scalar multiples of solutions are solutions. **Trivial solution** $y=0$ always works. **Linear Independence/Dependence:** Set $\{f_1,\dots,f_k\}$ is **linearly dependent** on $I$ if $\exists$ constants $c_i$ (not all zero) such that $c_1f_1 + \dots + c_kf_k = 0$ $\forall x\in I$. Otherwise, **independent**. - **Test:** Solve $c_1f_1 + \dots + c_kf_k = 0$ for each $c_i$. If **only** $c_i=0$ works as solutions, set is independent. **Fundamental Set of Solutions:** Linearly independent set of $n$ solutions to $L[y]=0$ on $I$. Any solution can be written as $y = c_1y_1 + \dots + c_ny_n$ ($c_{i}$ form basis for solution space). **Always exists** and has **$n$ elements** (matches ODE order). **Wronskian Test:** For $\{f_1,\dots,f_n\}$, define: $W = \begin{vmatrix} f_1 & \dots & f_n \\ f_1' & \dots & f_n' \\ \vdots & \ddots & \vdots \\ f_1^{(n-1)} & \dots & f_n^{(n-1)} \end{vmatrix}$ - **Independent** on $I$ $\iff$ $W \neq 0$ **for some** $x \in I$ (since $W$ is either **never zero** or **always zero** on $I$). Finding $\det$: (from top row) sum(products top left to bottom right) - sum(top right to bottom left), or along a row or column do $x\cdot \det(\text{remaining 2x2 matrix})$, remember alternating $+/-$ **Non-Homogeneous Solutions:** General solution: $y(x) = y_c(x) + y_p(x)$, where: - $y_c(x)$ = complementary (homogeneous) solution ($c_1y_1 + \dots + c_ny_n$). - $y_p(x)$ = **any** particular solution to $L[y]=g(x)$. **Superposition (Non-Homogeneous Case):** Given $L[y] = g_i(x)$ with particular solutions $y_i$, then: - $y = y_1 + \dots + y_k$ solves $L[y] = g_1(x) + \dots + g_k(x)$. - **Special case:** If $y_1, y_2$ solve $L[y]=g(x)$, then $y = 2y_1 - y_2$ **also solves** $L[y]=g(x)$ (since $L[2y_1 - y_2] = 2g(x) - g(x) = g(x)$). ### 3.3: Linear Equations with Constant Coefficients **First-Order Linear Homogeneous DE** **Form:** $L[y] = a \frac{dy}{dx} + by = 0$ **Solution:** $y = ce^{-rx}$, where $r = \frac{b}{a}$ and $c$ is arbitrary. **Higher-Order Linear Homogeneous ODEs** **Method:** Assume $y = e^{mx}$ (generalization of first-order case). **Auxiliary Equation:** Substitute to obtain an $n$-th order polynomial in $m$. Roots determine solutions. This creates a basis for the solutions of the homogenous equation. **Distinct Real Roots ($m_1 \neq m_2$):** $y(x) = C_1 e^{m_1 x} + C_2 e^{m_2 x}$ **Repeated Real Root ($m$):** $y(x) = C_1 e^{m x} + C_2 x e^{m x}$ **Complex Roots ($m = \alpha \pm \beta i$):** Use Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$. **Solution:** $y(x) = e^{\alpha x} \left( C_1 \cos(\beta x) + C_2 \sin(\beta x) \right)$ ### 3.4: Undetermined Coefficients #### 3.4.1: Linear ODE - Cauchy-Euler form? $\rightarrow$ Use that - Homogenous with constant coefficients? Use above. To find $x_p$: - Is $g(x)$ 'good' function? $\rightarrow$ undetermined coefficient. Otherwise, variation of parameters **Method of Undetermined Coefficients (for Nonhomogeneous ODEs)**. For if: - $L$ has **constant coefficients** and $g(x)$ is a **finite linear combination** of "good" functions: - **Constants**, **polynomials**, **exponentials**, **sines/cosines**. **General Approach** 1. **Solve the homogeneous equation** $L[y] = 0$ → Find $y_c(x)$ using 3.3. $\qquad\qquad y_p(x)$ based on $g(x)$ and its derivatives: - If $g(x) = P_n(x)e^{\alpha x}\cos(\beta x)$ or $P_n(x)e^{\alpha x}\sin(\beta x)$ ($P$ = polynomial), guess:$y_p = (A_0 + A_1x + \dots + A_nx^n)e^{\alpha x}\cos(\beta x) + (B_0 + B_1x + \dots + B_nx^n)e^{\alpha x}\sin(\beta x).$ - If $g(x)$ is a **polynomial**, guess a **general polynomial** of the same degree (eg. $Ax^{n} + Bx^{n-1}\dots + C$). - If $g(x)$ is an **exponential**, guess an **exponential** of the same form (eg. $Ae^{nx}$). - If $g(x)$ is trigonometric, guess both, $A\cos{nx}+B\sin{nx}$. - Careful for resonance! $Ce^{kx} \leftrightarrow r=k, C\text{cis}(kx) \leftrightarrow r=\pm ik, P_n(x)e^{\alpha x}\text{cis}{(\beta x)} \leftrightarrow \alpha \pm \beta i$, multiply by $x^{m}$, m is multiplicity of the root in the original equation. If RHS is a addition/subtraction combo, use linearity and find solutions for both individually - Resonance: If any term in $y_p$ **matches a term in $y_c$**, multiply $y_p$ by $x^n$, where $n$ is the smallest integer that removes the dupe. 2. **Plug $y_p$ into $L[y_p] = g(x)$** → Solve for coefficients $A, B, C, \dots$, match coefficients. $\qquad\qquad$ **General solution:** $y(x) = y_c(x) + y_p(x)$. - **IVP:** Given $y(x_0)$ and $y'(x_0)$ → Solve for constants in $y(x)$. $\qquad\qquad$**BVP:** Given $y(x_0)$ and $y(x_1)$ → Solve for constants in $y(x)$. **Summary**: Check if $L$ has constant coefficients. Solve $L[y] = 0$ → Find $y_c(x)$. If $g(x)$ is a "good" function → Undetermined coefficients. ### 3.5: Variation of Parameters **2nd-Order Linear ODE (General Form)** **Form:** $L[y] = y'' + P(x)y' + Q(x)y = f(x)$ - **Constant $P, Q$:** Homogeneous solution always possible. **Non-constant $P, Q$:** May not be solvable. **Complementary Solution:** $y_c(x) = c_1 y_1(x) + c_2 y_2(x)$ **Particular Solution ($y_p$):** Replace constants $c_1, c_2$ with functions $u_1(x), u_2(x)$ - Find Wronskian to eliminate higher derivatives, yielding:$\begin{cases}y_1 u_1' + y_2 u_2' = 0 \\y_1' u_1' + y_2' u_2' = f(x)\end{cases}$ - Cramer's rule: $\begin{align*} u_1(x) &= \int \frac{W_1(x)}{W(x)} \, dx \\ u_2(x) &= \int \frac{W_2(x)}{W(x)} \, dx \end{align*}$ - $W(x)$ = Wronskian of $\{y_1, y_2\}$. - $W_n(x)$ = Wronskian with $n$-th column replaced by $\begin{bmatrix} 0 \\ f(x) \end{bmatrix}$. - Substitute $u_{1},u_{2}$ back in $y_{p}(x)$ ### 3.6: Cauchy-Euler Equations **Cauchy-Euler Equation** **Form:** $L[y]=a_n x^n \frac{d^n y}{dx^n} + a_{n-1} x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1 x \frac{dy}{dx} + a_0 y = 0$, $x^{k}$ term matching order **Solution:** Assume $y = x^m$. Substitution yields **auxiliary equation**: $a_n m(m-1)\cdots(m-n+1) + \dots + a_1 m + a_0 = 0$ **Second-Order Case:** $L[y] = ax^2 y'' + bxy' + cy = 0$, **Auxiliary Equation:** $am(m-1) + bm + c = 0$ | **Case** | **General Solution ($y_c(x)$)** | | ---------------------------------------- | ------------------------------------------------------------------------- | | **Distinct Real Roots** ($m_1 \neq m_2$) | $c_1 x^{m_1} + c_2 x^{m_2}$ | | **Repeated Root** ($m$) | $c_1 x^m + c_2 x^m \log x$ | | **Complex Roots** ($\alpha \pm \beta i$) | $x^\alpha \left[ c_1 \cos(\beta \log x) + c_2 \sin(\beta \log x) \right]$ | ### 3.8: Linear Models for Initial Value Problems Gravity: $F_{g}=mg$ Spring force: $F_{s}=-k\Delta d$ Dynamical system for spring-mass system forced by gravity: $\ddot{x} + \omega^{2}x=0$, where $\omega=\sqrt{\frac{k}{m}}$ (natural frequency). Simple harmonic motion: $x(t)=c_{1}\cos{\omega t} + c_{2}\sin{\omega t}$, or this form: $x(t)=A\cos{(\omega t-\phi)}$ where $A=\sqrt{c_{1}^{2}+c_{2}^{2}}$ and $\phi=\arctan{\frac{c_{2}}{c_{1}}}$ Frictional force: $F_{f} = \beta \dot{x}$ where $\beta$ is damping constant. Damped oscillator ODE format: $x'' + 2\lambda x'+\omega^{2}x = 0$, no external force = homogeneous, external force = non-homogeneous, external force can be a function (forcing function). $\lambda$ is damping ratio $\frac{c}{m}$, $\omega$ is natural frequency, $\omega^{2}=\frac{k}{m}$.Three parts of solution: constant, transient ($x_{c}(t)$), steady state ($x_{p}(t)$). $A=\sqrt{C_{1}^{2}+C_{2}^{2}}, x(t)=C_{1} \cos{\omega t} + C_{2}\cos{\omega t}$. Deflection: $EI \frac{d^{4}y}{dx^{4}}=f(t)$, securely held means both $y$ and derivative are 0 at point, simply supported means derivative is nonzero ### 3.9: Linear Models: Boundary-Value Problems You need $n$ boundary conditions for a $n$ order DE. Review: if $A$ is $n \times n$ matrix, and $Av =\lambda v$, then $\lambda$ is an **eigenvalue** of $A$ and $v$ is an **eigenvector** of $A$. If $A$ has $n$ eigenvalues, we can express any vector as linear combination of eigenvalues of $A$. $L[y]=\lambda y$. If specify initial conditions $y(x_0) = y_0$, $y'(x_0) = y_1$ then, for any given $\lambda$, the eigenvalue problem has a unique solution on $(-\infty, \infty)$. However, if we specify boundary conditions $y(x_0) = y_0$, $y(x_1) = y_1$, $x_0 < x_1$, then the eigenvalue problem has a non-trivial solution on $[x_0, x_1]$ only for specific (but possibly infinitely many) values of $\lambda$. How to solve: - Set $L[y]=\lambda y$, three cases: $\lambda= 0, \lambda> 0, \lambda< 0$ - Evaluate $\lambda= 0$ by substituting in - Evaluate $\lambda> 0$ by performing $\lambda= -(\alpha^{2})$ - Evaluate $\lambda< 0$ by performing $\lambda= \beta^{2}$ - We do this to ease the eventual squaring of $\lambda$ since we know we will have a quadratic as our characteristic eq - Apply boundary conditions, try to find non-trivial solutions with homogeneous - If you find non-trivial solutions, substitute back ($-\alpha^{2}= \lambda, \beta^{2}=\lambda$) - Eigenvalue is the non-trivial solution, eigenfunctions are $y(x)$ the case you found the non-trivial solution with, with $\lambda$ subbed back in (in $\alpha$ or $\beta$ form if needed) - There will be another variable $n$, which represents some constant that creates a set of eigenfunctions - General solution is NOT eigenfunctions, solution is the linear combination of them $u(x,t)=\sum_{n=1}^{\infty}A_{n}u_{n}(x,t)$ - $A_{n}$ are coordinates of the solution in the eigenfunction basis, determined by initial conditions. - $u_{n}(x,t)$ are the eigenfunctions. - FOR 13.3: **Sturm-Liouville theory**: $A_n= \frac{2}{L}\int ^{L}_{0} f(x) \sin{\left( \frac{n \pi x}{L} \right)} \; dx$. We use this whole method with Sturm-Liouville equations with separation of variables, or when there is a $\lambda$ in the ODE ### 4.1: Laplace Transform **Laplace Transform**: Integral transform of $f(t)$ with kernel $e^{-st}$: $\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt$. To solve ODE with it, take transform of every term, substitute initial terms, solve algebraically for $Y(s)$, sometimes do partial fraction decomposition, then inverse transform back. Only for linear equations, IVPs, with constant coefficients. **Existence Conditions:** $f(t)$ piecewise continuous on $[0,\infty]$, exponential order, then exists on $s>c$ ($c$ is bounding exponential rate such that $f(t) \leq Me^{ct}$ on $t>T$ for constants $M,T$). $\mathcal{L}$ is a **linear operator**, so for constants $a, b$ and functions $f(t), g(t)$: $\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)$ ### 4.2: Inverse Laplace Transform and Transforms of Derivatives **Inverse Laplace Transform**: Linear but non-unique (unique up to null functions). $\mathcal{L}\{f'\} = sF(s)-f(0)$, $\mathcal{L}\{f''\} = s^2F(s)-sf(0)-f'(0)$, $\mathcal{L}\{f^{(n)}\} = s^nF(s) - \sum_{k=1}^n s^{n-k}f^{(k-1)}(0)$. **Note:** Requires ICs $f(0), f'(0), \dots, f^{(n-1)}(0)$. ### 4.3: Translation Theorems **First Translation (Frequency Shift):** $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ for $s>a+c$, $\quad\mathcal{L}^{-1}\{F(s-a)\} = e^{at}f(t)$ for $f(t)=\mathcal{L}^{-1}\left\{F(s)\right\}$. **Heaviside Step:** $\mathcal{U}(t-a) = \begin{cases} 0, & t t_0 + a. \end{cases}$ **Dirac delta function** is limit of impulses $\delta(t-t_{0})=\lim_{ a \to 0 }\delta _a(t-t_{a})$ - $\delta(t-t_0) = \begin{cases} \infty & t = t_0 \\ 0 & t \neq t_0 \end{cases}$ and $\int_{-\infty}^{\infty} \delta(t-t_0) \;dt = 1$ or $\int_{0}^{\infty} \delta(t-t_0) \;dt = 1$ if $t_0 \ge 0$ Sifting property of Dirac delta function (isolates $f(t)$ at translated point $a$): $f(t_0) = \int_0^\infty f(t)\delta(t-t_0)\;dt = \int_0^\infty f(t)\delta(t_0-t)\;dt = (f*\delta)(t_0)$ ### 13.1: Separable Partial Differential Equations General formula for linear second order PDE of two independent variables: $A(x,y)\frac{\partial^2 u}{\partial x^2} + B(x,y)\frac{\partial^2 u}{\partial x \partial y} + C(x,y)\frac{\partial^2 u}{\partial y^2} + D(x,y)\frac{\partial u}{\partial x} + E(x,y)\frac{\partial u}{\partial y} + F(x,y)u = G(x,y)$ PDEs are hard to solve unless they're separable, second order, and linear. If the PDE has constant coefficients: $B^{2}-4AC \quad \begin{cases} &>0 \quad &\text{hyperbolic} \\ & = 0 \quad & \text{parabolic} \\ &<0 \quad & \text{elliptic} \end{cases}$ Solving PDEs with dependent variable $u(x,t)$: - Separate variables, assume solution of form $u(x,t)=X(x)T{(t)}$, where $\frac{ \partial u }{ \partial t }=X'(x)T(t)$T - Move all $X$ terms and $T$ terms to separate sides, set both to separation variable $\lambda$ - The $T(t)$ side will be an ODE, the $X(x)$ side will be a BVP. Solve both - You will get eigenvalues and associated eigenfunctions from $X(x)$. - Use initial conditions $u(x,0)=f(x)=\sum_{n=1}^{\infty} A_{n}T(0)X_{n}(x)$ and appropriate dot product (integral) to find $A_{n}$ - Write down function as infinite eigenfunction series of form $u(x,t)=\sum_{n=1}^{\infty} A_{n}T(t)X_{n}(x)$ Need both boundary and initial conditions to solve constant coefficient separable PDE unless it's elliptic, then you only need boundary conditions ### 13.2: Classical PDEs: Boundary-Value Problems Wave equation: string of length $L$ vibrating, vertical displacement $u(x,t)$. If ends are clamped, boundary conditions are $u(0,t)=u(L,t)=0$, initial conditions are $u(x,0)=f(x)$ : $\frac{ \partial ^2u }{ \partial t^{2} } =a^{2}\frac{ \partial ^2u }{ \partial x^{2} }$ Heat equation: thin rod of cross-sectional area $A$, length $L$, thermal conductivity $k$, initial temperature distribution $f(x)$. Boundary conditions may include: both ends temperature = 0, heat flux conditions. Initial condition: $u(x,0)=f(x), 0