2.1 Algebra, Calculus & Differential Equations

Why math dominates the FE EE/CE

Mathematics is the largest single knowledge area on the FE Electrical and Computer exam at 11-17 of 110 questions. Just as important, math is the engine inside almost every other area: circuit analysis runs on complex algebra, linear systems and control systems run on differential equations and Laplace transforms, and signal processing runs on transforms and series. Strong, fast math fluency is the highest-leverage thing you can drill.

The exam is open to a searchable electronic copy of the NCEES FE Reference Handbook only, plus an NCEES-approved calculator (Casio FX-115, TI-36X, HP 33s/35s). Because every formula is in the Handbook, the tested skill is finding the right relationship fast and applying it cleanly — not recall. Ask one question first: is this a rate, an accumulation, a response curve, or a system of simultaneous equations?

Algebra, trig, and the calculus signal

Read the calculus notation as an engineering signal. A derivative is an instantaneous rate or slope; in circuits, i = C dv/dt and v = L di/dt. An integral is an accumulated quantity; charge q = ∫ i dt, energy w = ∫ p dt, and average value is the integral over an interval divided by the interval length. Trigonometry shows up in phasors and waveforms: keep the identities sin²θ + cos²θ = 1, sin(2θ) = 2 sinθ cosθ, and the law of cosines c² = a² + b² - 2ab cos C within reach.

A classic distractor: reporting the antiderivative instead of evaluating it at both bounds, or dropping the lower bound entirely.

Algebra fundamentals still earn points. Know the quadratic formula x = [-b ± sqrt(b² - 4ac)]/(2a) and what the discriminant b² - 4ac tells you: positive gives two real roots, zero a repeated root, negative a complex-conjugate pair. The same discriminant logic reappears in second-order circuit damping.

Differential equations under exam timing

Most FE ordinary differential equations (ODEs) are standard families you can recognize on sight. A first-order linear ODE has the form y' + a y = f(t). The source-free (homogeneous) case y' + a y = 0 gives an exponential response y = C e^(-at). In a series RC circuit the natural response decays with time constant τ = RC; in an RL circuit τ = L/R. After one time constant the quantity falls to about 36.8% of its initial value, and reaches roughly 99% of its final value after five time constants.

Worked example (first-order RC): A capacitor charged to 12 V discharges through R = 10 kΩ and C = 100 µF. Find v at t = 2 s. The time constant is τ = RC = (10×10³)(100×10⁻⁶) = 1 s, so v(t) = 12·e^(-t/τ) and v(2) = 12·e^(-2) = 12·0.1353 = 1.62 V. Notice you never integrated by hand: recognizing y' + ay = 0 gave the exponential form directly. The same template models RL current decay (τ = L/R), thermal cooling, and capacitor charging toward a source — only V₀ and the sign of the exponent change.

For a charging response toward a final value V_f, write v(t) = V_f + (V₀ - V_f)e^(-t/τ); the homogeneous (natural) part is the decaying exponential and the particular (forced) part is the steady-state constant V_f.

For second-order constant-coefficient ODEs, convert the equation to a characteristic equation and read the roots:

• Distinct real roots → two exponentials (overdamped).
• Repeated real root → add a t multiplier (critically damped).
• Complex-conjugate roots → decaying sine/cosine (underdamped), the ringing seen in series/parallel RLC step responses.

You rarely need a full derivation. Identify the root pattern and match it to the answer's functional form.

Linear algebra and vectors as shortcuts

Linear algebra appears whenever several unknowns must hold at once: mesh and nodal equations, regression normal equations, and state-space control models. For two equations, hand elimination is fastest. For three or more, the calculator matrix solver wins if your keystrokes are automatic. Eigenvalues come from det(A - λI) = 0; they set system modes and stability.

Worked example (eigenvalues): For A = [[4, 1], [2, 3]], the characteristic equation is (4-λ)(3-λ) - (1)(2) = λ² - 7λ + 10 = 0, giving λ = 5 and λ = 2. Check: trace 4+3 = 7 = 5+2, and determinant 12-2 = 10 = 5×2. Eigenvalues are the natural frequencies/modes of a linear system: in control work, eigenvalues with negative real parts mean a stable system, and a positive real part flags instability — the same root-location reasoning used for s-domain poles in Chapter 4.

Worked example (system of equations): Solve 2x + y = 5 and x - y = 1. Adding the equations eliminates y: 3x = 6, so x = 2 and y = 1. For two equations this hand elimination beats setting up a matrix; reserve the calculator solver for three or more unknowns.

Vectors add component-wise; the dot product A·B = |A||B|cosθ tests perpendicularity (zero when orthogonal), and the cross product magnitude |A||B|sinθ gives area and direction. Quick integrity checks before trusting a matrix answer:

1. Independent equations must equal the number of unknowns for a unique solution.
2. A determinant near zero signals dependency or ill conditioning.
3. For a 2x2, the trace equals the sum of eigenvalues and the determinant equals their product.
4. Units in every equation must be consistent.