WEEK 02. First Order Differential Equations

In this tutoring session, we skipped the planned review of geometry and statistics, as the tutees were already familiar with the material, and moved directly into first-order differential equations.

We began by introducing what differential equations are and their significance in engineering, using real-world examples to aid intuitive understanding.

The focus was on separable differential equations, where we derived general solutions and solved examples, including those with initial conditions.

We then covered linear first-order differential equations and introduced the integrating factor method as a key solution technique.

The session concluded with a Q&A to reinforce key concepts and highlight how differential equations apply to real engineering problems. 

WEEK 03. Second Order Differential Equations

In this tutoring session, we extended our exploration of differential equations to second-order cases, building on the foundation laid in the previous session on first-order equations.

The session began with an introduction to second-order differential equations, emphasizing their role in modeling acceleration or second-rate changes in physical systems. Real-world applications such as mass-spring-damper systems, RLC circuits, and heat transfer problems were used to contextualize their importance in engineering.

We focused primarily on linear second-order homogeneous differential equations with constant coefficients. The method of solving such equations using characteristic equations was introduced. We discussed how the nature of the roots of the characteristic equation—two distinct real roots, a repeated real root, or a pair of complex conjugates—determines the form of the general solution.

Example problems were worked through for each case, with special attention given to interpreting complex solutions as real-valued functions representing oscillatory behavior. The use of Euler's formula to convert complex exponentials into sine and cosine terms was explained to aid intuitive understanding.

We then moved on to initial value problems, where specific initial conditions are used to find a unique solution from the general one. The step-by-step process of applying initial values to solve for arbitrary constants was demonstrated, and we provided guidance to avoid common computational errors during this process.

The session wrapped up with a brief summary and a Q&A session, where we clarified conceptual misunderstandings and reviewed tricky calculation steps. Overall, the tutees gained a strong understanding of solving second-order linear differential equations and appreciated their relevance to real-world engineering problems.

WEEK 06&07. Fourier Series & Transform & Laplace Transform

In this tutoring session, we devoted approximately two hours to two main objectives: first, a conceptual and computational mastery of Fourier Series and Transforms for periodic and non-periodic signal representation; and second, a formal introduction to the Laplace Transform as a generalized tool for system analysis, bridging time-domain differential equations with frequency-domain methods. 

Session Overview

Fourier Series & Transform (First Hour):
We began by examining the fundamental idea that periodic functions can be represented as infinite sums of sinusoids, introducing the Fourier Series as a key tool in signal decomposition.

Tutees were guided through both the theoretical underpinnings and practical computations:

• Deriving Fourier coefficients a0,an,bna_0, a_n, b_na0​,an​,bn​ using orthogonality relationships.

• Exploiting function symmetry (even/odd/half-wave) to reduce computation.

• Graphical interpretation of harmonics and how they approximate original waveforms.

• Distinction between Fourier Series (periodic signals) and Fourier Transform (non-periodic signals) was emphasized to clear up common confusion.

To reinforce these ideas, we worked through problems involving piecewise functions such as square waves and sawtooth signals, guiding students in computing their spectral components and interpreting the resulting frequency-domain insights.

Common difficulties addressed included:

• Incorrect integral bounds or normalization constants.

• Misidentification of symmetry types leading to unnecessary computations.

• Confusion between real and complex exponential forms of the series.

Laplace Transform & System Analysis (Second Hour):
In the second half, we introduced the Laplace Transform as a more general tool than the Fourier Transform, particularly well-suited for solving differential equations involving initial conditions.

The session included:

• Definition of the Laplace Transform and key properties (linearity, time-shifting, differentiation).

• Step-by-step computation of Laplace pairs for common signals (step, exponential, sinusoidal).

• Introduction of transfer functions, enabling algebraic manipulation of differential systems.

• Emphasis on solving ordinary differential equations (ODEs) in the sss-domain and performing inverse transforms via partial fraction expansion.

Students practiced transforming differential equations to the Laplace domain, solving for outputs, and interpreting pole-zero configurations to infer system behavior.

Special care was taken to clarify:

• The difference between the sss-domain and the jωj\omegajω-axis (connection to Fourier).

• Common errors in using initial conditions incorrectly during transformation.

• Overreliance on formula memorization without structural understanding of system dynamics.

Wrap-Up and Q&A:
To close, we emphasized the unifying flow of the session—how the frequency-domain perspective (via Fourier and Laplace) allows for deeper insight and computational tractability in analyzing complex systems.