Separable differential equations

• Example: $\frac{dy}{dx} = x^2y$

Exact differential equations

• Example: $(2xy + 3)dx + (x^2 + 2y)dy = 0$

Linear differential equations

• Example: $\frac{dy}{dx} + p(x)y = q(x)$

• Example: $y’’ - 4y’ + 4y = 0$

• Example: $y’’ - 4y’ + 4y = x^2$

• Example: $y’’ - 4y’ + 4y = e^x$

• Example: Radioactive decay

• Example: Heating and cooling processes

• Example: Logistic growth model

• Example: Heat equation in one dimension

• Example: Wave equation in two dimensions

• Definition and properties

• Definition and properties

• Example: Solving second-order differential equations

• Example: Stability analysis in control systems

• $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$

• Example: Modeling the population of a city

• Example: Modeling the growth of a bacterial population

• Example: Rapid population growth in invasive species

• Example: LCR circuit

• Example: RC circuit with a charging capacitor

• Example: Signal processing and filtering

• $F = ma$

• Example: Simple harmonic motion

• Example: Motion of a falling object

• Example: Modeling the motion of a rocket

• Governing equations for fluid flow

• Example: Flow around an obstacle

• Example: Modeling weather patterns

• $F(s) = \mathcal{L}{f(t)} = \int_{0}^{\infty} e^{-st} f(t) dt$

• Example: $\mathcal{L}{2f(t) + 3g(t)} = 2F(s) + 3G(s)$

• Example: $\mathcal{L}{\frac{d^n}{dt^n} f(t)} = s^n F(s)$

• Example: Solving a second-order differential equation

• Example: Solving a differential equation with initial conditions

• Example: Analyzing the stability of a control system

• $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{2\pi nx}{L}\right) + b_n \sin\left(\frac{2\pi nx}{L}\right)\right)$

• Even and odd functions, periodic extension, convergence

• $F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x} dx$

• Linearity, time shifting, frequency shifting

• Example: Heat equation in one dimension

• Example: Filtering out specific frequencies

• Example: Denoising an audio signal

• $f(x, y, y’, y’’, \dots, y^{(n)}) \geq 0$

• Example: Boundedness, comparison, stability

• Example: Stability analysis in control systems

• Example: Stability of equilibrium solutions

• Example: Lyapunov function approach

• Example: Stability of a feedback control system

• $f(x, y, y’, y’’, \dots, y^{(n)}) \geq 0$

• Example: $\frac{dy}{dx} = x - y$

• The existence of an upper and lower bound
• The bounded derivative

• Example: Modeling the spread of a disease

• $g(x) \leq f(x, y, y’, y’’, \dots, y^{(n)}) \leq h(x)$

• Example: Comparing the solution with exponential decay

• Existence of upper and lower functions
• Monotonicity of the differential equation

• Example: Comparing the growth of different populations

• $f(x, y, y’, y’’, \dots, y^{(n)}) \leq 0$

• Example: Stability of a system of linear differential equations

• Lyapunov’s direct method
• Lyapunov’s indirect method

• Example: Stability of an inverted pendulum

• $y’ \geq y^2 - 2y$

• $e^x \leq y \leq e^{2x}$

• $y’ \leq -y$

• Example: Modeling the spread of an infectious disease

• Example: Stability of an electrical circuit

• Example: Stability of a predator-prey model

• Example: Determining optimal control strategies

• Example: Analyzing the stability of a chemical reaction

• Simplicity and generality of approach
• Provides qualitative information about solutions
• Useful for stability analysis

• Lack of quantitative information
• Difficulty in solving complex inequalities
• Applicability limited to certain types of problems