MAC2233 (6) - Calculus for Management Unit 9 review

Key Concepts

• Differential equations model relationships between a function and its derivatives
• Independent variable (usually denoted as $x$) represents the input
• Dependent variable (usually denoted as $y$) represents the output
• Order of a differential equation determined by the highest derivative present
  • First-order differential equations contain only first derivatives ($\frac{dy}{dx}$)
  • Second-order differential equations contain second derivatives ($\frac{d^2y}{dx^2}$)
• Initial conditions specify the value of the function and/or its derivatives at a specific point
• General solution of a differential equation contains arbitrary constants
• Particular solution obtained by applying initial conditions to the general solution

Types of Differential Equations

• Ordinary differential equations (ODEs) involve functions of a single independent variable
• Partial differential equations (PDEs) involve functions of multiple independent variables
• Linear differential equations have the dependent variable and its derivatives appearing linearly
  • Coefficients of the dependent variable and its derivatives are functions of only the independent variable
• Nonlinear differential equations have the dependent variable or its derivatives appearing nonlinearly
• Homogeneous differential equations have all terms containing the dependent variable or its derivatives
• Non-homogeneous (inhomogeneous) differential equations have terms without the dependent variable or its derivatives

Solving First-Order Differential Equations

• Separable equations can be written in the form $\frac{dy}{dx} = f(x)g(y)$
  • Separate variables and integrate both sides to find the general solution
• Linear first-order equations have the form $\frac{dy}{dx} + P(x)y = Q(x)$
  • Use an integrating factor to solve for the general solution
• Exact equations satisfy the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, where $M(x,y)dx + N(x,y)dy = 0$
  • Find a function $F(x,y)$ such that $\frac{\partial F}{\partial x} = M$ and $\frac{\partial F}{\partial y} = N$
• Bernoulli equations have the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $n \neq 0,1$
  • Substitute $v = y^{1-n}$ to transform the equation into a linear first-order equation

Applications in Business and Management

• Exponential growth and decay models (population growth, compound interest)
  • $\frac{dP}{dt} = kP$, where $P$ is the population size and $k$ is the growth rate
• Logistic growth model for limited resources (market saturation)
  • $\frac{dP}{dt} = kP(1-\frac{P}{L})$, where $L$ is the carrying capacity
• Continuously compounded interest (bank accounts, investments)
  • $\frac{dA}{dt} = rA$, where $A$ is the account balance and $r$ is the interest rate
• Inventory management and supply chain models (Economic Order Quantity)
  • $\frac{dI}{dt} = -D$, where $I$ is the inventory level and $D$ is the demand rate

Higher-Order Differential Equations

• Reduction of order technique for equations missing the dependent variable
  • Substitute $v = \frac{dy}{dx}$ to reduce the order of the equation
• Constant coefficient linear equations have the form $a_n\frac{d^ny}{dx^n} + a_{n-1}\frac{d^{n-1}y}{dx^{n-1}} + ... + a_1\frac{dy}{dx} + a_0y = f(x)$
  • Characteristic equation: $a_nr^n + a_{n-1}r^{n-1} + ... + a_1r + a_0 = 0$
  • General solution depends on the roots of the characteristic equation
• Method of undetermined coefficients for non-homogeneous equations with specific right-hand sides
  • Assume a particular solution based on the form of $f(x)$ with unknown coefficients
  • Substitute the assumed solution into the differential equation to solve for the coefficients

Systems of Differential Equations

• A system of differential equations consists of multiple equations involving multiple dependent variables
• First-order linear systems have the form $\frac{d\vec{x}}{dt} = A\vec{x} + \vec{f}(t)$, where $\vec{x}$ is a vector of dependent variables
  • Homogeneous systems ($\vec{f}(t) = \vec{0}$) have solutions that depend on the eigenvalues and eigenvectors of matrix $A$
  • Non-homogeneous systems can be solved using the method of variation of parameters
• Phase plane analysis for autonomous systems (right-hand side does not depend on $t$)
  • Equilibrium points: solutions where $\frac{d\vec{x}}{dt} = \vec{0}$
  • Linearization near equilibrium points to determine stability

Numerical Methods and Technology

• Euler's method for approximating solutions
  • $y_{n+1} = y_n + h\frac{dy}{dx}|_{x=x_n}$, where $h$ is the step size
• Runge-Kutta methods (RK4) for higher accuracy
  • Weighted average of slope estimates at multiple points within each step
• Adaptive step size methods (Dormand-Prince) for error control
  • Adjust the step size based on the estimated local truncation error
• Software tools for solving and visualizing differential equations (MATLAB, Python libraries)
  • ode45 function in MATLAB for solving ODEs
  • scipy.integrate module in Python for numerical integration and ODE solvers

Practice Problems and Examples

• Solve the separable equation: $\frac{dy}{dx} = xy^2$
  • Separate variables and integrate: $\int \frac{1}{y^2}dy = \int x dx$
• Solve the linear first-order equation: $\frac{dy}{dx} + 2y = e^x$
  • Multiply both sides by the integrating factor $e^{\int 2dx} = e^{2x}$
• Solve the second-order constant coefficient equation: $y'' - 5y' + 6y = 0$
  • Characteristic equation: $r^2 - 5r + 6 = 0$, roots: $r = 2, 3$
  • General solution: $y = c_1e^{2x} + c_2e^{3x}$
• Model the spread of a rumor in a population of 1000 people, where the rate of spread is proportional to the product of the number of people who have heard the rumor and those who haven't
  • Let $y(t)$ be the number of people who have heard the rumor at time $t$
  • $\frac{dy}{dt} = ky(1000-y)$, solve using the method for separable equations