Engineering Mathematics 2: practice problems and exam questions

The second mathematics course for engineers: second-order differential equations, matrices and vectors, integration techniques with volumes of revolution, partial derivatives and double integrals, series and Taylor polynomials, the Laplace transform and convolution. These are the tools that circuits, vibrations and control courses build on.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Differential equationsexponential growth and decay · Newton's law of cooling · first-order linearcalculations
  2. Matrices & determinants2×2 determinant · 3×3 and products · eigenvalues and systemscalculations
  3. Vectors: dot & cross productmagnitude and components · dot product · cross product and momentscalculations
  4. Integralspolynomials · exponentials, logs and substitution · areas between curvescalculations
  5. Multivariable calculuspartial derivatives and double integrals · gradient, tangent plane and chain rule · critical points and integrals over regionscalculations
  6. Sequences & seriesarithmetic and geometric series · convergence and power series · Taylor and Maclaurin polynomialscalculations
  7. Laplace transformstable pairs and value theorems · inverse transform by partial fractions · ODEs and transfer functionscalculations
  8. Convolution & LTI systemsdiscrete convolution · continuous convolution · impulse response and Fourier seriescalculations

Exam tips for Engineering Mathematics 2

  • For a linear ODE, write the characteristic equation first: its roots decide the form of the solution.
  • Choose u in integration by parts as the factor that becomes simpler when you differentiate it.
  • Use a Laplace table and partial fractions rather than integrating the transform by hand.

Worked examples

Differential equationsEasy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0

Find r

Show hints
  1. Substitute y = ert.
  2. r² + 4r + 3 = 0.
  3. Factor it or use the quadratic formula.
Show the answer and the step-by-step solution

Second-order linear equations: for y″ + b·y′ + c·y = 0 try y = ert: this gives the characteristic equation r² + b·r + c = 0. Two real roots r₁ ≠ r₂ give y = A·er₁t + B·er₂t (overdamped); a double root r gives y = (A + Bt)·ert (critically damped); complex roots −α ± jω give y = e−αt(A cos ωt + B sin ωt) (underdamped, oscillating). The initial values y(0) and y′(0) fix A and B. Written as y″ + 2ζωny′ + ωn²y = 0, ωn is the natural frequency and ζ the damping ratio: ζ < 1 oscillates. With a constant input, y″ + by′ + cy = c·K, the solution is the steady value K plus the free response.

Characteristic equation

r² + 4r + 3 = (r + 1)(r + 3) = 0

Roots

r = −1, r = −3
closest to zero: r = −1

Answer

r = −1

The general solution is y = A·e−1t + B·e−3t.

Matrices & determinantsMedium
A =−153−5−1−3−513

Find the determinant det A of the 3×3 matrix A shown.

A = [−1 5 3; −5 −1 −3; −5 1 3]

Find det A

Show hints
  1. Expand along the first row with signs + − +.
  2. Each term is an entry times the 2×2 determinant left when you cross out its row and column.
  3. minors: 0, −30, −10.
Show the answer and the step-by-step solution

Matrices: the determinant of a 2×2 matrix is ad − bc. For 3×3, expand along a row: each entry times the determinant of the 2×2 matrix left after removing its row and column, with alternating signs + − +. Eigenvalues λ solve det(A − λI) = 0; for 2×2 that is λ² − (trace)·λ + det = 0.

2×2 minors of the first row

M₁₁ = 0, M₁₂ = −30, M₁₃ = −10

Cofactor expansion

det A = (−1)·0 − 5·(−30) + 3·(−10) = 120

Answer

det A = 120

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Inverse Laplace transform calculator (partial fractions)Laplace transform calculatorMatrix determinant and inverse calculatorEigenvalue and eigenvector calculatorHalf-life and exponential decay calculatorSystem of linear equations solver (2×2 and 3×3)Integral calculator (definite and indefinite)

Related courses

Engineering MathematicsCalculus 2Numerical MethodsAP Calculus practice problemsDifferential EquationsSignals and SystemsCalculus 1Statics
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.