Ordinary Differential Equations Vocabulary

Overview of Differential Equations

• Definition: A differential equation is an equation where the unknown is a function, and both the function and its derivatives may appear in the equation.

• Differential equations are essential for physical theories, including:

  • Newton’s and Lagrange’s equations (Classical Mechanics).

  • Maxwell’s equations (Electromagnetism).

  • Schrödinger’s equation (Quantum Mechanics).

  • Einstein’s equation (General Relativity).

• Ordinary Differential Equation (ODE): The unknown function depends on a single independent variable, usually represented as $t$.

  • Examples:

    • Radioactive Decay: $\frac{du}{dt}(t) = -k u(t)$, where k > 0 is a constant.

    • Newton's Law: $m \frac{d^2 x}{dt^2}(t) = f(t, x(t), \frac{dx}{dt}(t))$.

• Partial Differential Equation (PDE): The unknown function depends on two or more independent variables (e.g., $t, x, y, z$).

  • Examples:

    • Heat Equation: $\frac{\partial T}{\partial t}(t, x) = k (\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2})$.

    • Wave Equation: $\frac{\partial^2 u}{\partial t^2} = v^2 (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2})$.

• Order of an Equation: The highest derivative order appearing in the equation.

First Order Linear Equations

• Definition: A first order ODE is linear iff the source function $f$ is linear in its second argument:

  • $y'(t) = a(t)y + b(t)$.

• Constant Coefficients: Linear iff both $a$ and $b$ are constants.

• Variable Coefficients: Linear iff $a$ or $b$ depend on $t$.

• General Solution (Constant Coefficients): For $y' = ay + b$ with $a \neq 0$:

  • $y(t) = c e^{at} - \frac{b}{a}$, where $c \in \mathbb{R}$.

• Integrating Factor Method:

  • Transformation into a total derivative: $\mu(t)y' - a\mu(t)y = \mu(t)b(t)$.

  • The integrating factor $\mu(t)$ is chosen such that $\mu'(t) = -a(t)\mu(t)$, resulting in $\mu(t) = e^{-A(t)}$ where $A(t) = \int a(t) dt$.

  • Final form: $(\mu y)' = \mu b$.

• Initial Value Problem (IVP): Consists of the ODE and an initial condition $y(t_0) = y_0$.

  • Unique Solution for Constant Coefficients: $y(t) = (y_0 + \frac{b}{a}) e^{a(t-t_0)} - \frac{b}{a}$.

Linear Variable Coefficient Equations and Bernoulli Equations

• Theorem 1.2.1 (Variable Coefficients): If $a, b$ are continuous on $(t_1, t_2)$, then $y' = a(t)y + b(t)$ has fundamental solution:

  • $y(t) = c e^{A(t)} + e^{A(t)} \int e^{-A(t)}b(t) dt$

  • Where $A(t) = \int a(t) dt$.

• Bernoulli Equation: A first order nonlinear equation of the form:

  • $y' = p(t)y + q(t)y^n$, where $n \in \mathbb{R}$.

  • Transformation: For $n \neq 1$, define $v = \frac{1}{y^{(n-1)}}$.

  • Linearized Equation for $v$: $v' = -(n-1)p(t)v - (n-1)q(t)$.

Separable and Euler Homogeneous Equations

• Separable Equations: Defined as $h(y)y' = g(t)$.

  • Solution via direct integration: $\int h(y) dy = \int g(t) dt + c$.

  • Implicit vs Explicit form: Solutions are often roots of algebraic equations (e.g., cubic polynomials).

• Euler Homogeneous Equations: Equations of the form $y' = F(\frac{y}{t})$.

  • Scale Invariance: $F(\frac{cy}{ct}) = F(\frac{y}{t})$.

  • Solving Strategy: Substitute $v = \frac{y}{t}$. This yields $y = tv$ and $y' = v + tv'$.

  • Resulting Separable Equation: $v' = \frac{F(v)-v}{t}$, or $\frac{v'}{F(v)-v} = \frac{1}{t}$.

Exact Differential Equations

• Definition: An equation $N(t, y)y' + M(t, y) = 0$ is exact iff $\partial_t N = \partial_y M$.

• Potential Function (Poincaré Theorem): If exact, there exists $\psi(t, y)$ such that:

  • $\partial_y \psi = N$ and $\partial_t \psi = M$.

  • The differential equation behaves as $\frac{d\psi}{dt}(t, y(t)) = 0$.

  • The solutions satisfy the level curve equation $\psi(t, y(t)) = c$.

• Semi-Exact Equations: Equations that are not exact but become exact after multiplying by an integrating factor $\mu$.

  • If $h = \frac{\partial_y M - \partial_t N}{N}$ depends only on $t$, then $\mu(t) = e^{H(t)}$ where $H = \int h dt$.

  • If $\ell = -\frac{\partial_y M - \partial_x N}{M}$ depends only on $y$, then $\mu(y) = e^{L(y)}$ where $L = \int \ell dy$.

Applications of Linear Equations

• Exponential Decay: $\frac{dN}{dt} = -kN$.

  • Solution: $N(t) = N_0 e^{-kt}$.

  • Half-life (\tau): The time for half the material to decay. Relationship: $k\tau = \ln(2)$.

• Newton’s Cooling Law: The rate of change of temperature is proportional to the difference between the object and the medium temperature.

  • $\Delta T' = -k \Delta T$, where $\Delta T(t) = T(t) - T_s$.

  • Solution: $T(t) = (T_0 - T_s) e^{-kt} + T_s$.

• Mixing Problems:

  • Volume balance: $V'(t) = r_i - r_o$.

  • Salt balance: $Q'(t) = r_i q_i(t) - r_o \frac{Q(t)}{V(t)}$.

  • Model: $Q'(t) = a(t)Q(t) + b(t)$.

Nonlinear Equations: Picard-Lindelöf and Picard Iteration

• Picard-Lindelöf Theorem: Concerns the IVP $y' = f(t, y), y(t_0) = y_0$.

  • If $f$ is continuous and Lipschitz continuous in $y$ on a domain $D_a$, there exists a unique solution in a neighborhood of $t_0$.

• Picard Iteration: Constructs a sequence of approximate solutions:

  • $y_0(t) = y_0$.

  • $y_{n+1}(t) = y_0 + \int_{t_0}^t f(s, y_n(s)) ds$.

• Existence vs Uniqueness: Non-uniqueness can occur if the Lipschitz condition is not met (e.g., $y' = y^{1/3}$ has multiple solutions for $y(0) = 0$).

• Direction Fields: Graphical representation of $\text{slope} = f(t, y)$, used for qualitative analysis.

Second Order Linear Equations

• General Form: $y'' + a_1(t) y' + a_0(t) y = b(t)$.

• Linearity and Superposition: If $y_1, y_2$ solve the homogeneous equation ($b(t)=0$), then $c_1 y_1 + c_2 y_2$ also solves it.

• Wronskian: $W_{12}(t) = y_1 y'_2 - y'_1 y_2$.

  • $y_1, y_2$ are linearly independent iff $W_{12} \neq 0$.

• Abel’s Theorem: The Wronskian of two solutions satisfies $W' + a_1(t) W = 0$, so $W(t) = W(t_0) e^{-\int a_1(s) ds}$.

• Homogeneous Constant Coefficients: Characteristic polynomial $p(r) = r^2 + a_1 r + a_0 = 0$.

  • Roots $r_1, r_2$ distinct real: $y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t}$.

  • Roots repeated real ($r_0$): $y(t) = (c_1 + c_2 t) e^{r_0 t}$.

  • Roots complex ($\alpha \pm i\beta$): Real fundamental solutions are $e^{\alpha t}\cos(\beta t)$ and $e^{\alpha t}\sin(\beta t)$.

• Reduction of Order: If $y_1$ is known, $y_2(t) = y_1(t) \int \frac{e^{-A_1(t)}}{y_1^2(t)} dt$.

• Euler Equidimensional Equation: $(t - t_0)^2 y'' + a_1 (t - t_0) y' + a_0 y = 0$.

  • Indicial polynomial: $r(r-1) + a_1 r + a_0 = 0$.

  • Solutions involve $(t-t_0)^r$ or $(t-t_0)^r \ln(t-t_0)$.

• Nonhomogeneous Equations:

  • Undetermined Coefficients: Guess $y_p$ based on the shape of the source term $b(t)$.

  • Variation of Parameters: $y_p = u_1 y_1 + u_2 y_2$.

    • $u'_1 = -\frac{y_2 f}{W_{12}}$, $u'_2 = \frac{y_1 f}{W_{12}}$.

Power Series Solutions

• Regular Points: Points where coefficients $p(x), q(x)$ are analytic.

• Method: Assume $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n$. Plug into the ODE and solve the recurrence relation for coefficients $a_n$.

• Legendre Equation: $(1-x^2)y'' - 2xy' + \ell(\ell+1)y = 0$. Solutions lead to Legendre Polynomials $P_n(x)$.

• Regular Singular Points: Point $x_0$ is a regular singular point iff $(x-x_0) p(x)$ and $(x-x_0)^2 q(x)$ are analytic at $x_0$.

• Frobenius Method: Assume $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^{n+r}$.

  • Define the indicial equation to find possible values for $r$.

  • Bessel Equation: $x^2 y'' + xy' + (x^2 - \alpha^2)y = 0$. Solution defines Bessel functions $J_\alpha(x)$.

Laplace Transform Method

• Definition: $L[f](s) = \int_0^\infty e^{-st} f(t) dt$.

• Key Properties:

  • Linearity: $L[af + bg] = aL[f] + bL[g]$.

  • Translation (I): $L[u(t-c) f(t-c)] = e^{-cs} L[f]$.

  • Translation (II): $L[e^{ct} f(t)] = L[f](s-c)$.

  • Derivative to Multiplication: $L[f'] = s L[f] - f(0)$.

  • Higher Derivatives: $L[f''] = s^2 L[f] - s f(0) - f'(0)$.

• Common Transforms:

  • $L[1] = \frac{1}{s}$

  • $L[e^{at}] = \frac{1}{s-a}$

  • $L[t^n] = \frac{n!}{s^{n+1}}$

  • $L[\sin(at)] = \frac{a}{s^2+a^2}$, $L[\cos(at)] = \frac{s}{s^2+a^2}$.

• Dirac Delta Functions (Generalized Sources): $\delta(t-c)$.

  • $L[\delta(t-c)] = e^{-cs}$ for $c \geq 0$.

  • Integral property: $\int_a^b f(t) \delta(t-c) dt = f(c)$.

• Convolution: $(f*g)(t) = \int_0^t f(\tau) g(t-\tau) d\tau$.

  • Theorem: $L[f*g] = L[f]L[g]$.

Systems of Linear Differential Equations

• Representation: $\mathbf{x}' = A(t)\mathbf{x} + \mathbf{b}(t)$.

• General Solution (Homogeneous): $\mathbf{x}_{gen}(t) = \sum c_i \mathbf{x}^{(i)}(t)$, where $\mathbf{x}^{(i)}$ are fundamental solutions.

• Matrix Exponential Solution: For constant matrix $A$, IVP $\mathbf{x}' = A\mathbf{x}, \mathbf{x}(t_0) = \mathbf{x}_0$:

  • $\mathbf{x}(t) = e^{A(t-t_0)} \mathbf{x}_0$.

• Diagonalizable Case: If $A = PDP^{-1}$, then $e^{At} = P e^{Dt} P^{-1}$. Solution decomposes into eigenmodes: $\mathbf{x}(t) = \sum c_i e^{\lambda_i t} \mathbf{v}^{(i)}$.

• Fundamental Matrix ($X(t)$): Matrix formed by fundamental solution vectors.

• Nonhomogeneous Systems: $\mathbf{x}(t) = e^{A(t-t_0)} \mathbf{x}_0 + \int_{t_0}^t e^{A(t-\tau)} \mathbf{b}(\tau) d\tau$.

Autonomous Systems and Stability

• Definition: System $y' = f(y)$ where $f$ is time-independent.

• Critical Points: $\mathbf{x}_c$ such that $f(\mathbf{x}_c) = 0$.

• Stability Analysis:

  • Attractor (Sink): Solutions flow toward it.

  • Repeller (Source): Solutions flow away.

  • Linearization (Jackobian Matrix): $D\mathbf{f}(\mathbf{x}_c)$. Stability governed by eigenvalues.

• Phase Portraits (2D):

  • Nodes (real distinct eigenvalues, same sign).

  • SADDLE points (real distinct eigenvalues, opposite signs).

  • Spirals/Centers (complex eigenvalues).

• Competition Species Model (Lotka-Volterra):

  • $x'_1 = x_1(3 - x_1 - 2x_2)$, $x'_2 = x_2(2 - x_2 - x_1)$.

  • Basins of attraction determine which species survives.

Boundary Value Problems and Heat Equation

• Eigenfunction Problems: Find $\lambda$ and non-zero $y$ for $y'' + \lambda y = 0$ with BC like $y(0)=0, y(L)=0$.

  • Solutions: Eigenvalues $\lambda_n = (\frac{n\pi}{L})^2$, Eigenfunctions $y_n(x) = \sin(\frac{n\pi x}{L})$.

• Fourier Series: Expands $f(x)$ on $[-L, L]$ as $\frac{a_0}{2} + \sum [a_n \cos(\frac{n\pi x}{L}) + b_n \sin(\frac{n\pi x}{L})]$.

• Heat Equation Application: $\partial_t u = k \partial^2_x u$.

  • Separation of Variables: $u(t, x) = v(t) w(x)$.

  • Result for Dirichlet BC ($u(t,0)=u(t,L)=0$): $u(t, x) = \sum c_n e^{-k(\frac{n\pi}{L})^2 t} \sin(\frac{n\pi x}{L})$ where $c_n$ are Fourier coefficients of initial data $f(x)$.