📊Mathematical Modeling Unit 6 – Differential Equations

What Are Differential Equations?

• Equations involving derivatives of one or more dependent variables with respect to one or more independent variables
• Describe the rate of change of a quantity in relation to another quantity
• Arise in various fields of science, engineering, and economics when modeling real-world phenomena
• Classified based on the order (highest derivative), linearity, and whether the coefficients are constants or functions
• Denoted by the order of the highest derivative, such as first-order, second-order, or nth-order differential equations
• Solutions to differential equations are functions that satisfy the equation and any given initial or boundary conditions
  • Initial conditions specify the value of the function and/or its derivatives at a particular point
  • Boundary conditions specify the value of the function and/or its derivatives at the endpoints of an interval

Types of Differential Equations

• Ordinary Differential Equations (ODEs) involve derivatives with respect to a single independent variable
  • Example: $\frac{dy}{dx} = x^2 + y$
• Partial Differential Equations (PDEs) involve partial derivatives with respect to multiple independent variables
  • Example: $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ (heat equation)
• Linear differential equations have the dependent variable and its derivatives appearing linearly, with coefficients being constants or functions of the independent variable(s)
  • Example: $y'' + 2y' + y = 0$
• Nonlinear differential equations have the dependent variable or its derivatives appearing in a nonlinear manner
  • Example: $y' = y^2 + x$
• Homogeneous differential equations have all terms containing the dependent variable and its derivatives
  • Example: $y'' + y = 0$
• Non-homogeneous differential equations have at least one term that is a function of the independent variable(s) only
  • Example: $y'' + y = \sin(x)$

Solving First-Order Differential Equations

• Separation of variables method involves separating the variables and integrating both sides
  • Applicable when the equation can be written as $\frac{dy}{dx} = f(x)g(y)$
  • Steps: separate variables, integrate both sides, and solve for the dependent variable
• Integrating factor method is used for linear first-order equations of the form $y' + P(x)y = Q(x)$
  • Multiply both sides by an integrating factor $\mu(x) = e^{\int P(x)dx}$ to make the left-hand side a total derivative
  • Integrate both sides and solve for the dependent variable
• Exact differential equations are of the form $M(x, y)dx + N(x, y)dy = 0$, where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
  • Find a function $F(x, y)$ such that $\frac{\partial F}{\partial x} = M(x, y)$ and $\frac{\partial F}{\partial y} = N(x, y)$
  • Set $F(x, y) = C$ and solve for the dependent variable
• Bernoulli equations are of the form $y' + P(x)y = Q(x)y^n$, where $n \neq 0, 1$
  • Substitute $v = y^{1-n}$ to transform the equation into a linear first-order equation
  • Solve using the integrating factor method

Higher-Order Differential Equations

• Reduction of order method is used when one solution to a linear homogeneous equation is known
  • Substitute $y = v(x)y_1(x)$, where $y_1(x)$ is the known solution, and solve for $v(x)$
• Method of undetermined coefficients is used for non-homogeneous linear equations with specific right-hand side terms (polynomials, exponentials, sines, or cosines)
  • Assume a particular solution with unknown coefficients and determine the coefficients by substituting the solution into the equation
• Variation of parameters method is a general method for solving non-homogeneous linear equations
  • Find the general solution to the corresponding homogeneous equation
  • Substitute the constants with functions and solve for these functions using a system of equations
• Laplace transform method converts a differential equation into an algebraic equation in the frequency domain
  • Take the Laplace transform of the equation and initial conditions
  • Solve the resulting algebraic equation for the transformed dependent variable
  • Apply the inverse Laplace transform to obtain the solution in the time domain

Applications in Mathematical Modeling

• Population dynamics models describe the growth or decline of populations over time
  • Example: logistic growth model $\frac{dP}{dt} = rP(1 - \frac{P}{K})$, where $P$ is the population size, $r$ is the growth rate, and $K$ is the carrying capacity
• Mechanical systems, such as spring-mass systems or pendulums, can be modeled using second-order differential equations
  • Example: simple harmonic motion $m\frac{d^2x}{dt^2} + kx = 0$, where $m$ is the mass and $k$ is the spring constant
• Heat transfer and diffusion processes are modeled using partial differential equations
  • Example: heat equation $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$, where $u$ is the temperature and $\alpha$ is the thermal diffusivity
• Fluid dynamics and aerodynamics problems involve partial differential equations, such as the Navier-Stokes equations
  • Example: continuity equation $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$, where $\rho$ is the fluid density and $\vec{v}$ is the velocity vector
• Chemical reactions and kinetics can be modeled using systems of first-order differential equations
  • Example: first-order reaction $\frac{d[A]}{dt} = -k[A]$, where $[A]$ is the concentration of reactant $A$ and $k$ is the rate constant

Key Theorems and Concepts

• Existence and uniqueness theorem states that a first-order initial value problem $y' = f(x, y)$, $y(x_0) = y_0$ has a unique solution if $f$ and $\frac{\partial f}{\partial y}$ are continuous in a rectangle containing $(x_0, y_0)$
• Fundamental set of solutions for a linear homogeneous equation is a set of linearly independent solutions that can be used to generate all solutions
  • The number of solutions in the fundamental set is equal to the order of the equation
• Wronskian is a determinant used to determine the linear independence of solutions
  • For functions $y_1(x), y_2(x), \ldots, y_n(x)$, the Wronskian is $W(y_1, y_2, \ldots, y_n) = \det \begin{vmatrix} y_1 & y_2 & \ldots & y_n \\ y_1' & y_2' & \ldots & y_n' \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \ldots & y_n^{(n-1)} \end{vmatrix}$
  • If the Wronskian is non-zero at a point, the solutions are linearly independent
• Superposition principle states that the general solution to a linear non-homogeneous equation is the sum of the general solution to the corresponding homogeneous equation and a particular solution to the non-homogeneous equation

Common Pitfalls and How to Avoid Them

• Forgetting to include the constant of integration when solving differential equations
  • Always add a constant of integration after indefinite integration and determine its value using initial or boundary conditions
• Incorrectly applying the chain rule when separating variables or using the integrating factor method
  • Be careful when differentiating composite functions and use the chain rule correctly
• Misidentifying the type of differential equation or the appropriate solution method
  • Analyze the equation carefully to determine its type (linear, nonlinear, homogeneous, non-homogeneous) and select the appropriate solution method
• Errors in algebraic manipulation or integration when solving differential equations
  • Double-check algebraic steps and use integration techniques correctly
  • Verify that the solution satisfies the original differential equation
• Misinterpreting or misapplying initial or boundary conditions
  • Ensure that the initial or boundary conditions are used correctly when determining constants of integration or particular solutions
• Overlooking the possibility of multiple solutions or solution families
  • Consider the possibility of multiple solutions, especially for nonlinear equations or equations with arbitrary constants
  • Verify that all solutions are found and that they are linearly independent

Practice Problems and Solutions

1. Solve the first-order differential equation $\frac{dy}{dx} = x^2 + y^2$ using the separation of variables method.

   • Solution:
     • Separate variables: $\frac{dy}{1 + y^2} = x^2 dx$
     • Integrate both sides: $\arctan(y) = \frac{x^3}{3} + C$
     • Solve for $y$: $y = \tan(\frac{x^3}{3} + C)$

2. Find the general solution to the second-order linear homogeneous equation $y'' - 5y' + 6y = 0$.

   • Solution:
     • Characteristic equation: $r^2 - 5r + 6 = 0$
     • Roots: $r_1 = 2$, $r_2 = 3$
     • General solution: $y = c_1e^{2x} + c_2e^{3x}$, where $c_1$ and $c_2$ are arbitrary constants

3. Use the method of undetermined coefficients to find a particular solution to the non-homogeneous equation $y'' + 4y = 3\sin(2x)$.

   • Solution:
     • Assume a particular solution of the form $y_p = A\cos(2x) + B\sin(2x)$
     • Substitute $y_p$ into the equation and solve for $A$ and $B$
     • Particular solution: $y_p = \frac{3}{5}\sin(2x)$

4. Apply the Laplace transform to solve the initial value problem $y'' + 4y' + 3y = 0$, $y(0) = 1$, $y'(0) = 0$.

   • Solution:
     • Take the Laplace transform of the equation and initial conditions
     • Solve the resulting algebraic equation for $Y(s)$: $Y(s) = \frac{s + 4}{s^2 + 4s + 3}$
     • Perform partial fraction decomposition and apply the inverse Laplace transform
     • Solution: $y = e^{-x}(\cos(\sqrt{2}x) + \frac{1}{\sqrt{2}}\sin(\sqrt{2}x))$

5. Model the population growth of a species with a carrying capacity of 1000 and a growth rate of 0.2 using the logistic growth model. Find the population size after 10 years if the initial population is 100.

   • Solution:
     • Logistic growth model: $\frac{dP}{dt} = 0.2P(1 - \frac{P}{1000})$
     • Separate variables and integrate: $\int \frac{dP}{P(1 - \frac{P}{1000})} = \int 0.2 dt$
     • Solve for $P(t)$: $P(t) = \frac{1000}{1 + 9e^{-0.2t}}$
     • Substitute $t = 10$ and $P(0) = 100$: $P(10) \approx 993$