🪝Ordinary Differential Equations Unit 3 – First-Order DE Applications

Key Concepts

• First-order differential equations involve the first derivative of a function and can model various real-world phenomena
• The general form of a first-order DE is $\frac{dy}{dt} = f(t, y)$, where $f$ is a function of the independent variable $t$ and the dependent variable $y$
• Initial conditions specify the value of the dependent variable at a specific point, allowing for a unique solution
• Separable equations can be written in the form $\frac{dy}{dt} = g(t)h(y)$, where $g(t)$ is a function of $t$ and $h(y)$ is a function of $y$
  • Separable equations can be solved by separating the variables and integrating both sides
• Linear equations have the form $\frac{dy}{dt} + P(t)y = Q(t)$, where $P(t)$ and $Q(t)$ are functions of $t$
  • The integrating factor method is used to solve linear equations
• Exact equations satisfy the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, where $M(x, y)$ and $N(x, y)$ are the coefficients of the equation $M(x, y)dx + N(x, y)dy = 0$
• Steady-state solutions represent the long-term behavior of a system, where the dependent variable remains constant over time

Types of First-Order DEs

• Separable equations can be separated into two functions, one depending on $t$ and the other on $y$
• Linear equations involve the dependent variable $y$ and its first derivative, with coefficients that are functions of $t$
• Exact equations have the form $M(x, y)dx + N(x, y)dy = 0$, where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
• Homogeneous equations can be transformed into separable equations by substituting $y = vx$
• Bernoulli equations have the form $\frac{dy}{dt} + P(t)y = Q(t)y^n$, where $n$ is a real number different from 0 and 1
  • Bernoulli equations can be transformed into linear equations by substituting $z = y^{1-n}$
• Riccati equations have the form $\frac{dy}{dt} = P(t)y^2 + Q(t)y + R(t)$, where $P(t)$, $Q(t)$, and $R(t)$ are functions of $t$
• Clairaut's equations have the form $y = xy' + f(y')$, where $f$ is a differentiable function

Modeling with First-Order DEs

• Population growth can be modeled using the equation $\frac{dP}{dt} = kP$, where $P$ is the population size and $k$ is the growth rate
  • The logistic equation $\frac{dP}{dt} = kP(1 - \frac{P}{K})$ accounts for carrying capacity $K$, limiting population growth
• Radioactive decay follows the equation $\frac{dN}{dt} = -\lambda N$, where $N$ is the number of atoms and $\lambda$ is the decay constant
• Newton's law of cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature
  • The equation $\frac{dT}{dt} = -k(T - T_a)$ models this phenomenon, where $T$ is the object's temperature, $T_a$ is the ambient temperature, and $k$ is a positive constant
• Mixing problems involve substances entering and leaving a tank or container, with the concentration changing over time
• Electrical circuits with resistors, capacitors, and inductors can be modeled using first-order DEs
  • The equation $\frac{dI}{dt} + \frac{R}{L}I = \frac{V}{L}$ describes the current $I$ in an RL circuit with resistance $R$, inductance $L$, and voltage $V$

Solution Methods

• Separation of variables involves rewriting the equation in the form $\frac{dy}{dt} = g(t)h(y)$ and integrating both sides
  • The variables are separated, and the equation becomes $\int \frac{1}{h(y)}dy = \int g(t)dt$
• The integrating factor method is used to solve linear equations of the form $\frac{dy}{dt} + P(t)y = Q(t)$
  • The integrating factor is $\mu(t) = e^{\int P(t)dt}$, and the solution is $y = \frac{1}{\mu(t)}\left(\int \mu(t)Q(t)dt + C\right)$
• Exact equations can be solved by finding a function $F(x, y)$ such that $\frac{\partial F}{\partial x} = M(x, y)$ and $\frac{\partial F}{\partial y} = N(x, y)$
  • The solution is implicitly given by $F(x, y) = C$, where $C$ is an arbitrary constant
• Numerical methods, such as Euler's method and Runge-Kutta methods, approximate solutions using iterative techniques
  • Euler's method uses the formula $y_{n+1} = y_n + hf(t_n, y_n)$, where $h$ is the step size and $f(t, y)$ is the right-hand side of the DE
• Laplace transforms convert a DE into an algebraic equation, which can be solved for the transformed function and then inverted to obtain the solution
  • The Laplace transform of $\frac{dy}{dt}$ is $sY(s) - y(0)$, where $Y(s)$ is the Laplace transform of $y(t)$

Real-World Applications

• Population dynamics, including the growth of bacteria, animal populations, and human populations
  • The logistic equation is used to model population growth with limited resources (carrying capacity)
• Radioactive decay in nuclear physics and radiometric dating
  • Carbon-14 dating is based on the exponential decay of the radioactive isotope $^{14}C$
• Heat transfer and Newton's law of cooling in thermodynamics
  • The cooling of a hot object in a cooler environment (coffee cup, metal forging)
• Mixing problems in chemistry and environmental science
  • Pollutants entering and leaving a lake or river, or the concentration of a drug in the bloodstream
• Electrical circuits with resistors, capacitors, and inductors in electrical engineering
  • The charging and discharging of a capacitor in an RC circuit
• Simple harmonic motion in physics and engineering
  • The motion of a mass attached to a spring or a pendulum
• Fluid dynamics and the flow of fluids through pipes or channels
  • The Hagen-Poiseuille equation describes the pressure drop in a fluid flowing through a cylindrical pipe

Common Challenges

• Identifying the type of first-order DE and selecting the appropriate solution method
• Correctly separating variables and integrating both sides of the equation
• Determining the integrating factor for linear equations and applying it correctly
• Verifying that an equation is exact and finding the function $F(x, y)$ that satisfies the partial derivative conditions
• Applying initial conditions to determine the value of the arbitrary constant in the general solution
• Interpreting the results of the solution in the context of the real-world problem being modeled
• Dealing with equations that do not fit into standard categories (separable, linear, exact) and require creative transformations or substitutions
• Understanding the limitations and assumptions of the models used to describe real-world phenomena

Practice Problems

• Solve the separable equation $\frac{dy}{dt} = t^2y$ with the initial condition $y(1) = 2$
• Find the general solution of the linear equation $\frac{dy}{dt} + 2ty = t^2$ and determine the particular solution that satisfies $y(0) = 1$
• Verify that the equation $(2x + y)dx + (x - y)dy = 0$ is exact and find the solution that passes through the point $(1, 1)$
• Use Euler's method with a step size of $h = 0.1$ to approximate the solution of $\frac{dy}{dt} = t - y$ with $y(0) = 1$ over the interval $[0, 1]$
• A tank initially contains 100 L of water with 5 kg of salt dissolved in it. Water containing 0.1 kg/L of salt enters the tank at a rate of 2 L/min, and the well-mixed solution leaves the tank at the same rate. Find the amount of salt in the tank after 20 minutes
• The population of a city is growing according to the logistic equation $\frac{dP}{dt} = 0.05P(1 - \frac{P}{100,000})$. If the initial population is 10,000, find the population after 10 years
• An RC circuit has a resistance of 10 kΩ and a capacitance of 100 μF. If the initial charge on the capacitor is 10 mC, find the charge on the capacitor after 0.1 seconds

Further Resources

• "Elementary Differential Equations and Boundary Value Problems" by Boyce and DiPrima
  • A comprehensive textbook covering first-order DEs, higher-order DEs, and boundary value problems
• "Differential Equations with Applications and Historical Notes" by Simmons and Krantz
  • Emphasizes the connection between DEs and their applications, with historical context and examples
• Paul's Online Math Notes (https://tutorial.math.lamar.edu/Classes/DE/DE.aspx)
  • A free online resource with detailed explanations, examples, and practice problems for differential equations
• MIT OpenCourseWare: Differential Equations (https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/)
  • Lecture notes, assignments, and exams from MIT's undergraduate differential equations course
• Wolfram MathWorld: Differential Equations (https://mathworld.wolfram.com/topics/DifferentialEquations.html)
  • A collection of articles and resources on various types of differential equations and their properties
• MATLAB and Mathematica documentation for solving differential equations
  • Both software packages have built-in functions and tutorials for numerically solving and visualizing DEs
• "Nonlinear Dynamics and Chaos" by Strogatz
  • An introduction to nonlinear systems and chaos theory, with applications to physics, biology, and engineering