ECE 205: Advanced Calculus 1 for Electrical and Computer Engineers

Sources and References

Equivalent UW courses — AMATH 250 (Introduction to Differential Equations), MATH 211 (Advanced Calculus 1 for Honours Math), MATH 228 (Differential Equations for Physics / Chemistry) Primary textbook — William E. Boyce, Richard C. DiPrima, and Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 11th ed., Wiley, 2017. Supplementary references — R. Kent Nagle, Edward B. Saff, and Arthur David Snider, Fundamentals of Differential Equations, 9th ed., Pearson, 2017; Richard Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th ed., Pearson, 2012.

Equivalent UW Courses

ECE 205 is the engineering ODE / transforms course, and its closest single equivalent is AMATH 250, which covers first- and second-order ODEs, Laplace transforms, and a short introduction to systems. MATH 211 is the honours-math version of the same material, taught with more rigour (existence and uniqueness proofs, careful discussion of linear operators). MATH 228 is the Physics / Chemistry variant with an applied flavour similar to ECE 205. None of AMATH 250, MATH 211, or MATH 228 covers Fourier series and the Fourier transform — in the Math Faculty those live in AMATH 353 or AMATH 351 / 353. So ECE 205 is really AMATH 250 plus a compressed slice of AMATH 353.

What This Course Adds Beyond the Equivalents

• Fourier series and Fourier transform in the same course as ODEs. This is specifically the content AMATH 250 / MATH 211 leave out.
• Intro to PDEs via separation of variables. The heat and wave equations on a finite interval are solved using Fourier series in the last two weeks — content normally postponed to AMATH 353 / AMATH 353 in the Math Faculty.
• Linear-system and circuit motivation. Laplace transforms are presented as the tool of choice for solving LCCODEs arising in electrical circuits, with transfer functions and impulse response framed ahead of schedule.
• Omits, relative to AMATH 250 / MATH 211: systems of ODEs with matrix exponentials, phase-plane analysis, nonlinear stability, series solutions near singular points, and rigorous Picard existence / uniqueness proofs. Relative to AMATH 353: convergence of Fourier series, Parseval, and Sturm-Liouville theory.

Topic Summary

First-Order ODEs

Separable, linear (integrating factor), and exact equations. Direction fields for qualitative behaviour. Engineering examples: RC and RL circuits, Newton’s law of cooling.

Second-Order Linear ODEs with Constant Coefficients

Homogeneous equations via the characteristic polynomial, with real-distinct, repeated, and complex-conjugate root cases. Particular solutions by undetermined coefficients and variation of parameters. The canonical damped-oscillator ODE

\[ m\ddot x + c\dot x + kx = F(t) \]

is used to introduce overdamped, critically damped, and underdamped regimes.

Applications: Mechanical and Electrical Oscillators

RLC circuits as the electrical analogue of the spring-mass-damper. Resonance, beating, and the frequency response at forced sinusoidal input.

The Laplace Transform

Definition \( \mathcal L\{f\}(s) = \int_0^\infty e^{-st}f(t)\,dt \), region of convergence, linearity, and shift theorems. Transforms of derivatives, step functions, and the Dirac delta. Solving IVPs by algebraic manipulation in the \( s \)-domain, followed by partial-fraction inverse transforms. Transfer functions and impulse response for LTI systems.

Fourier Series

Expansion of a periodic function \( f(t) \) of period \( 2L \) as

\[ f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n\cos\frac{n\pi t}{L} + b_n\sin\frac{n\pi t}{L} \right) \]

Euler formulas for the coefficients; even / odd extensions on a half-interval. Basic discussion of pointwise convergence and the Gibbs phenomenon at jumps.

Fourier Transform

The continuous-frequency transform as a limit of Fourier series as \( L \to \infty \). Linearity, duality, and the derivative / modulation properties. Convolution theorem and its interpretation as filtering.

Partial Differential Equations via Separation of Variables

The 1D heat equation \( u_t = \alpha u_{xx} \) and wave equation \( u_{tt} = c^2 u_{xx} \) on a finite rod / string with Dirichlet or Neumann boundary conditions. Separation yields an eigenvalue problem; the solution is assembled as a Fourier series matching the initial data.