Differential Equations for the FE Exam
Differential equations are among the most feared topics on the FE exam, but the good news is that the exam tests a relatively narrow set of differential equation types and solution methods. If you focus your study on the most commonly tested patterns, you can handle these questions confidently.
What Is a Differential Equation?
A differential equation is an equation that relates a function to its derivatives. In engineering, differential equations model dynamic systems: motion, heat transfer, fluid flow, electrical circuits, and population growth, among others. The solution to a differential equation is a function (not a number) that satisfies the equation.
Classifying Differential Equations
Before solving, classify the equation:
- Order: The highest derivative present (first-order has dy/dx, second-order has d2y/dx2)
- Linear vs. nonlinear: Linear equations have the dependent variable and its derivatives appearing to the first power only
- Homogeneous vs. nonhomogeneous: Homogeneous equations equal zero; nonhomogeneous equations have a forcing function on the right side
- Ordinary vs. partial: ODEs involve one independent variable; PDEs involve multiple (the FE exam focuses on ODEs)
First-Order Ordinary Differential Equations
Separable Equations
A separable ODE can be written as dy/dx = f(x)*g(y). To solve:
- Separate variables: dy/g(y) = f(x)dx
- Integrate both sides
- Solve for y if possible
- Apply initial conditions to find the constant of integration
Example: dy/dx = xy can be separated as dy/y = x dx. Integrating gives ln|y| = x^2/2 + C, so y = Ae^(x^2/2).
First-Order Linear Equations
A first-order linear ODE has the form dy/dx + P(x)y = Q(x). To solve, use an integrating factor:
- Calculate the integrating factor: mu(x) = e^(integral of P(x) dx)
- Multiply both sides of the equation by mu(x)
- The left side becomes d/dx[mu(x)*y]
- Integrate both sides and solve for y
Second-Order Linear ODEs with Constant Coefficients
These are the most commonly tested differential equations on the FE exam. The general form is: ay'' + by' + cy = f(x).
Homogeneous Case: ay'' + by' + cy = 0
Solve the characteristic equation: ar^2 + br + c = 0. The solution depends on the roots:
- Two distinct real roots (r1, r2): y = C1*e^(r1*x) + C2*e^(r2*x)
- One repeated real root (r): y = (C1 + C2*x)*e^(r*x)
- Complex conjugate roots (alpha +/- beta*i): y = e^(alpha*x) * [C1*cos(beta*x) + C2*sin(beta*x)]
This is the pattern you must memorize. It appears frequently on the FE exam.
Nonhomogeneous Case: ay'' + by' + cy = f(x)
The general solution is: y = y_h + y_p, where y_h is the homogeneous solution (found above) and y_p is a particular solution. Common methods for finding y_p include:
- Method of undetermined coefficients: Guess the form of y_p based on f(x) and solve for the coefficients. This works when f(x) is a polynomial, exponential, sine, cosine, or combination thereof.
- Variation of parameters: A more general method that works for any f(x), but is more computationally intensive.
Engineering Applications
Recognizing which physical systems lead to which types of differential equations helps you set up and solve problems faster:
- Exponential growth/decay: dy/dt = ky leads to y = y0*e^(kt). Applies to radioactive decay, population growth, and Newton's law of cooling.
- Spring-mass-damper systems: mx'' + cx' + kx = F(t). The characteristic equation roots tell you whether the system is overdamped, underdamped, or critically damped.
- RC and RL circuits: First-order linear ODEs that model charging and discharging of capacitors and inductors.
- RLC circuits: Second-order linear ODEs analogous to spring-mass-damper systems.
Initial Value Problems
Most FE exam differential equation problems are initial value problems (IVPs), where you are given the value of y (and possibly y') at a specific point. Use these initial conditions to determine the constants in your general solution.
Study Tips for Differential Equations
- Focus on the characteristic equation method for second-order ODEs; it appears most frequently
- Practice classifying equations before solving; choosing the wrong method wastes time
- Know the three cases for characteristic equation roots (distinct real, repeated, complex) cold
- Practice setting up differential equations from word problems, not just solving given equations
- Use the reference handbook for solution templates when available
Differential equations on the FE exam are very manageable once you master the standard solution techniques. Focus on recognizing equation types and applying the correct method, and practice enough problems to build speed and confidence.