Ordinary Differential Equations: Lectures by Arman Bayramyan

Definition and Scope of Ordinary Differential Equations

• Mathematical Definition: An Ordinary Differential Equation (ODE) is an equation of the form:     $F(x, y, y', \dots, y^{(n)}) = 0$     where:

  • $x \in \mathbb{R}$ is the independent variable.

  • $y = y(x)$ is the unknown function.

  • $F$ is a function of $n+2$ variables ($n \ge 1$).

• Terminology:

  • Differential: The equation contains derivatives of the unknown function.

  • Ordinary: The derivatives are "ordinary" (with respect to a single variable) as opposed to partial derivatives.

• Examples:

  • ODE: $y(y')^2 - 2x^2y = \sin(x)$. Here $n = 1$, and the function is $F(x_1, x_2, x_3) = x_2x_3^2 - 2x_1^2x_2 - \sin(x_1)$.

  • Non-ODE: $y' = y(y(x))$ is not a differential equation because it cannot be represented in the standard form (it is a functional differential equation).

Real-World Applications of Differential Equations

• Problem of a Falling Body:

  • Consider a body of mass $m$ in free fall from height $h$ without initial velocity.

  • Let $x(t)$ be the height and $v(t) = \frac{dx}{dt}$ be the velocity at time $t$.

  • According to Newton's Second Law: $m \mathbf{a} = m \frac{d\mathbf{v}}{dt} = \mathbf{F}$.

  • If gravity is the only force: $m \frac{d^2x}{dt^2} = -mg$.

  • If air resistance is proportional to velocity ($\mathbf{r}(t) = -\gamma \mathbf{v}(t)$, where $\gamma > 0$):         $m \frac{d^2x}{dt^2} = -mg - \gamma \frac{dx}{dt}$         $m \frac{dv}{dt} = -mg - \gamma v$

• Radiocarbon Dating Method:

  • Used to determine the age of organic materials based on the amount of radioactive Carbon-14 ($^{14}C$).

  • The rate of decay is proportional to the mass of the substance at that moment.

  • Let $c(t)$ be the concentration at time $t$ ($t=0$ at death):         $\frac{dc}{dt} = -kc$         where $k > 0$.

• Predator-Prey Model (Volterra Model):

  • Let $x(t)$ be the number of prey (rabbits) and $y(t)$ the number of predators (wolves).

  • Model proposed by Vito Volterra:

    1. In the absence of predators, prey grows proportionally: $\frac{dx}{dt} = \alpha x$.

    2. Meetings between species are proportional to $xy$, leading to prey loss: $\frac{dx}{dt} = \alpha x - \beta xy$.

    3. Predators die without food but grow through meetings: $\frac{dy}{dt} = -\gamma y + \delta xy$.

  • Represented as a system:         $\begin{cases} \frac{dx}{dt} = \alpha x - \beta xy \\ \frac{dy}{dt} = -\gamma y + \delta xy \end{cases}$

Classification of Differential Equations

• Ordinary vs. Partial: ODEs involve one variable; Partial Differential Equations (PDEs) involve functions of multiple variables (e.g., Wave Equation: $\frac{\partial^2u(x, t)}{\partial t^2} = c^2 \frac{\partial^2u(x, t)}{\partial x^2}$).

• Linearity: An ODE is linear if $F$ is linear with respect to variables $y, y', \dots, y^{(n)}$. General form:     $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_0(x)y = f(x)$

  • Non-linear example: The mathematical pendulum equation $\frac{d^2\theta}{dt^2} + \frac{g}{L} \sin(\theta) = 0$

• Order: The order of the highest derivative present in the equation.

• Normal Form: The highest-order derivative is explicitly expressed by the remaining terms:     $y^{(n)} = f(x, y, y', \dots, y^{(n-1)})$

Solutions and the Cauchy Problem

• Definition of Solution (Integral): A function $u: I \rightarrow \mathbb{R}$ defined on interval $I \subset \mathbb{R}$ such that:

  1. $u \in C^n(I)$.

  2. $F(x, u(x), u'(x), \dots, u^{(n)}(x)) = 0$ for all $x \in I$.

• General Solution: For a simple equation like $\frac{dy}{dx} = f(x)$, the general solution is $y(x) = \int f(x)dx + C$. The graph of a solution is called an integral curve.

• Implicit Solution: A relation $\Phi(x, y) = 0$ that defines at least one function $y = u(x)$ solving the ODE.

• The Cauchy Problem: An ODE coupled with initial conditions (e.g., $y(x_0) = y_0$).

• Existence and Uniqueness Theorems:

  • Peano Theorem: If $f(x, y)$ is continuous in rectangle $\Omega$, there exists at least one solution to the Cauchy problem in the interval $[x_0 - h, x_0 + h]$, where $h = \min\{a, \frac{b}{M}\}$ and $M = \max |f(x, y)|$.

  • Picard Theorem: If $f(x, y)$ is continuous and satisfies the Lipschitz condition ($|f(x, y_1) - f(x, y_2)| \le L|y_1 - y_2|$) with respect to the second variable, there is a unique solution in $[x_0 - h, x_0 + h]$.

Direction Fields and Isoclines

• Direction Field: The set of vectors $(1, f(x, y))$ placed at every point $(x, y)$ in the domain, representing slopes of integral curves.

• Isocline: A line where the slope of the direction field is constant (level curves of $f(x, y)$).

First-Order Equations: Separable and Linear

• Separable Equations: $y' = p(x)q(y)$.

  • Method: Divide by $q(y)$ and integrate: $\int \frac{1}{q(y)}dy = \int p(x)dx + C$.

  • Stationary Solutions: Constant solutions where $q(y) = 0$.

• Linear Equations: $y' + a(x)y = b(x)$.

  • Homogeneous Case ($b(x) \equiv 0$): $y(x) = Ce^{-\int a(x)dx}$.

  • Non-homogeneous Case: General solution $y(x) = e^{-\int a(x)dx} \left( C + \int b(x) e^{\int a(x)dx} dx \right)$, or $y = y_h + y_p$.

  • Variation of Constants Method: Replace constant $C$ with a function $C(x)$ to find the particular solution.

Exact Equations and Integrating Factors

• Exact Equation: $M(x, y)dx + N(x, y)dy = 0$ is exact if there exists a potential function $F(x, y)$ such that $\frac{\partial F}{\partial x} = M$ and $\frac{\partial F}{\partial y} = N$.

• Condition for Exactness: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ in a simply connected domain.

• Solution: $F(x, y) = C$. A specific potential formula is:     $F(x, y) = \int_{x_0}^x M(s, y_0)ds + \int_{y_0}^y N(x, t)dt$

• Integrating Factor: If the equation is not exact, multiply by $\mu(x, y)$ such that $(\mu M)_y = (\mu N)_x$. The factor must satisfy:     $M \mu_y - N \mu_x = (N_x - M_y)\mu$

Substitution Methods

• Homogeneous Equations: $M(x, y)dx + N(x, y)dy = 0$ where $M, N$ are homogeneous functions of the same degree. Substitute $u = \frac{y}{x}$. The equation becomes $x \frac{du}{dx} = F(u) - u$.

• Bernoulli Equation: $y' + P(x)y = Q(x)y^\alpha$ ($\alpha \neq 0, 1$). Substitute $z = y^{1-\alpha}$ to transform it into a linear equation.

• Riccati Equation: $y' = P(x) + Q(x)y + R(x)y^2$. If one particular solution $y_1(x)$ is known, substitute $y = y_1 + z$ to get a Bernoulli equation for $z$.

Numerical Methods for First-Order ODEs

• Euler's Method: Approximates the integral curve with a sequence of tangents.

  • Step: $h = \frac{b - x_0}{n}$.

  • Iterative formula: $y_{k+1} = y_k + hf(x_k, y_k)$.

  • Local Error: $O(h^2)$, Global Error: $O(h)$.

• Runge-Kutta Method (RK4): Uses a weighted average of four slopes per step.

  • $k_{i1} = f(x_i, y_i)$

  • $k_{i2} = f(x_i + \frac{h}{2}, y_i + \frac{h}{2} k_{i1})$

  • $k_{i3} = f(x_i + \frac{h}{2}, y_i + \frac{h}{2} k_{i2})$

  • $k_{i4} = f(x_i + h, y_i + h k_{i3})$

  • $y_{i+1} = y_i + \frac{h}{6} (k_{i1} + 2k_{i2} + 2k_{i3} + k_{i4})$

  • Global Error: $O(h^4)$.

Second-Order Linear Equations

• Spring-Mass Motion: Developed from Newton's law and Hooke's Law ($F = -k(s+x)$). Equation: $m \frac{d^2x}{dt^2} = -kx$.

• General Form: $y'' + P(x)y' + Q(x)y = R(x)$.

• Existence/Uniqueness: With two conditions $y(x_0) = y_0$ and $y'(x_0) = y'_0$, a unique solution exists if $P, Q, R$ are continuous in $(a, b)$.

• Homogeneous General Solution: $y = c_1 y_1 + c_2 y_2$.

• Wronskian: $W[y_1, y_2] = y_1 y_2' - y_2 y_1'$. If $W \ne 0$, then $y_1, y_2$ are linearly independent.

• Abel-Liouville Formula: $W(x) = W(x_0)e^{-\int_{x_0}^x P(t)dt}$.

• Constant Coefficients: $ay'' + by' + cy = 0$. Solve characteristic equation $ar^2 + br + c = 0$.

  1. Distinct Real Roots: $y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}$.

  2. Complex Roots ($r = \lambda \pm \mu i$): $y(x) = e^{\lambda x}(c_1 \cos(\mu x) + c_2 \sin(\mu x))$.

  3. Repeated Root ($r_1 = r_2 = r$): $y(x) = (c_1 + c_2 x)e^{rx}$.

• Non-homogeneous General Solution: $y = y_h + y_p$. Use Variation of Constants to find $y_p$.

Series Solutions and the Frobenius Method

• Analytical Function: Function representable by a power series around $x_0$.

• Ordinary Point: $x_0$ where $P(x)$ and $Q(x)$ are analytical.

• Series Expansion: $\sum_{n=0}^\infty a_n (x-x_0)^n$. Determine coefficients using recurrence relations.

• Regular Singular Point: $x_0$ where $(x-x_0)P(x)$ and $(x-x_0)^2Q(x)$ are analytical.

• Frobenius Method: Solution of form $y(x) = x^r \sum_{n=0}^\infty a_n x^n$.

Higher-Order Linear ODEs

• General Theory: $y^{(n)} + p_1(x)y^{(n-1)} + \dots + p_n(x)y = f(x)$.

• Operator Notation: $Ly = [D^n + p_1 D^{n-1} + \dots + p_n I](y)$. Operator $L$ is linear.

• Fundamental System: A set of $n$ linearly independent solutions $y_1, \dots, y_n$. Independence verified if the $n \times n$ Wronskian determinant is non-zero.

• Constant Coefficients: Roots of $P(r) = a_0 r^n + a_1 r^{n-1} + \dots + a_n = 0$.

  • For an $m$-multiplicity root $r_1$, solutions are $e^{r_1 x}, x e^{r_1 x}, \dots, x^{m-1} e^{r_1 x}$.

Systems of Linear ODEs

• Normal Form of Systems:     $\begin{cases} y'_1 = A_{11}(x)y_1 + \dots + A_{1n}(x)y_n + f_1(x) \\ \dots \\ y'_n = A_{n1}(x)y_1 + \dots + A_{nn}(x)y_n + f_n(x) \end{cases}$

• Vector Notation: $\mathbf{y}' = \mathbf{A}(x)\mathbf{y} + \mathbf{f}(x)$.

• Fundamental Matrix $\Psi(x)$: A matrix whose columns are linearly independent vector solutions.

• Matrix Exponent: For constant $\mathbf{A}$, the fundamental matrix is $e^{\mathbf{A}x} = \sum_{m=0}^\infty \frac{(\mathbf{A}x)^m}{m!}$.

  • Calculation: Use Jordan Normal Form $A = PJP^{-1}$. Then $e^A = Pe^JP^{-1}$.

Autonomous Systems and Stability

• Autonomous System: Independent of explicit time $t$: $\mathbf{y}' = \mathbf{F}(\mathbf{y})$.

• Equilibrium/Critical Point: Point $\mathbf{y}^*$ such that $\mathbf{F}(\mathbf{y}^*) = \mathbf{0}$.

• Phase Portrait: Visualization of trajectories in the state space.

• Classification of 2D Linear Critical Points:

  • Stable Node: Real, distinct, negative eigenvalues.

  • Unstable Node: Real, distinct, positive eigenvalues.

  • Saddle Point: Real eigenvalues of opposite signs.

  • Center: Purely imaginary eigenvalues.

  • Stable/Unstable Focus: Complex eigenvalues with negative/positive real parts.

• Stability Definitions:

  1. Stable: Solutions starting near $\mathbf{y}^*$ stay near forever.

  2. Asymptotically Stable: Solutions starting near $\mathbf{y}^*$ approach it as $t \rightarrow \infty$.

• Lyapunov Stability for Almost Linear Systems: If the Jacobian $J = \mathbf{F}'(\mathbf{y}^*)$ has eigenvalues with non-zero real parts, the stability of the non-linear system matches the linearized system $y' = Jy$.