AE 223: Differential Equations and Balance Laws

Sources and References

Equivalent UW courses — AMATH 250 (Intro to Differential Equations), MATH 211 (Advanced Calculus 1 for Honours Math), MATH 215 (Linear ODEs), AMATH 351 (ODEs 2) Primary textbook — Lebl, Jiří. Notes on Diffy Qs: Differential Equations for Engineers. Lulu, 2022. Supplementary references — Trench, William F. Elementary Differential Equations with Boundary Value Problems. Brooks/Cole (freely distributable edition).

Equivalent UW Courses

For Honours Math students, the closest analogue to AE 223 is AMATH 250, which covers essentially the same menu of first- and second-order ODE techniques, Laplace transforms, and a brief glimpse of PDEs and Fourier series. MATH 215 restricts itself to linear ODE theory (existence, uniqueness, linear independence, Wronskians) at a more theoretical level, while AMATH 351 extends AMATH 250 into series solutions, phase plane analysis, stability, and Sturm-Liouville problems. MATH 211 overlaps only in its vector calculus and balance-law flavor (gradient, divergence, line and surface integrals) rather than in ODE technique. AE 223 compresses the AMATH 250 syllabus and pairs it with engineering balance-law motivation.

What This Course Adds Beyond the Equivalents

AE 223 is framed around engineering balance laws — mass, momentum, and energy conservation — so every solution technique is immediately cashed out as a model of a mixing tank, heat exchanger, spring-mass-damper, or transmission-line-like PDE. It puts more weight on numerical methods (Euler’s method) earlier than AMATH 250 typically does, and it spends meaningful time on higher-order ODEs and applications rather than on theoretical questions of existence and uniqueness.

What it omits relative to the math-faculty stream: the rigorous treatment of linear operator theory in MATH 215, the series solutions and phase-plane material of AMATH 351, and the Sturm-Liouville / eigenfunction framing of boundary-value problems. PDEs are introduced only as separation-of-variables exercises on the heat and wave equations rather than as a general theory.

Topic Summary

First-order ODEs

Covers general and particular solutions, initial and boundary conditions, separable equations, linear equations via integrating factors, exact equations, and autonomous equations. Trench chapters 2.1-2.6 and Lebl 1.1-1.6, 1.8 are the reference entries. The engineering motivation is mass-balance problems on well-mixed tanks and first-order RC-like systems.

Numerical methods

Euler’s method is introduced for initial value problems where a closed-form solution is unavailable or inconvenient. Error behavior is discussed informally — no Runge-Kutta, no formal convergence analysis — corresponding to Trench 3.1 and Lebl 1.7.

Applications of linear first-order equations

A full week is spent translating first-order ODEs into engineering balance-law models: compartment models, Newton’s law of cooling, simple population dynamics, and RL circuits (Trench chapter 4).

Second-order linear ODEs

Homogeneous and non-homogeneous constant-coefficient equations, the characteristic equation, method of undetermined coefficients, reduction of order, and variation of parameters. Covers Trench 5.1-5.7 and Lebl 2.1-2.2. Physical applications include spring-mass-damper systems, LRC circuits, and forced resonance.

Higher-order ODEs and applications

Extension of second-order technique to higher-order linear constant-coefficient equations, with emphasis on engineering applications rather than general theory (Trench 6.1-6.2, 9.1-9.4; Lebl 2.4, 2.6).

Laplace transforms

Definition, transforms of elementary functions, shifting theorems, transforms of derivatives, inverse transforms, piecewise forcing functions, impulse and Dirac delta inputs, and convolution. Two weeks on Trench 8.1-8.7 and Lebl 6.1-6.5 and 4.6. This is the workhorse technique for the course’s circuits-and-systems-flavored problems.

Partial differential equations and Fourier series

Brief introduction to separation of variables on the heat, wave, and Laplace equations, with Fourier-series expansion of initial data. Covers Trench 11.1-11.3 and Lebl 4.2-4.4. The goal is exposure, not a general theory: students should be able to set up and solve a canonical separation-of-variables PDE on a bounded interval.

Applications of PDEs

Closing week (Trench 12.1) applies the separation-of-variables machinery to a concrete engineering boundary-value problem, typically heat conduction in a rod or membrane vibration.