Questions: Systems of First-Order Linear Differential Equations

Question 1

In the system y' = Ay + b(t), what does the matrix A encode?

• A. The initial conditions of the system
• B. How each variable contributes to the rate of change of the others
• C. The solutions of the individual equations
• D. The number of independent variables

Answer: (B)

Each entry A_ij captures how the j-th variable contributes to the rate of change of the i-th variable. This coupling between equations is exactly what distinguishes a system from a collection of independent scalar ODEs, and it is why scalar methods must be extended using linear algebra.

Question 2

A single second-order ODE y'' = f(t, y, y') can always be converted into a system of two first-order ODEs.

• T. True
• F. False

Answer: True

Setting y₁ = y and y₂ = y' transforms the equation into the system y₁' = y₂ and y₂' = f(t, y₁, y₂). This reduction is why first-order systems are the fundamental object of study — every higher-order ODE reduces to one, so methods for systems apply universally.

Question 3

Why is matrix form y' = Ay useful for solving systems of linear ODEs?

Model answer: Matrix form unifies the system into a single equation structurally identical to the scalar case y' = ay, enabling eigenvalue decomposition to find solutions of the form e^(λt)v, where λ is an eigenvalue of A and v is the corresponding eigenvector.

The scalar ODE y' = ay has solution y = Ce^(at). For the matrix system, the role of a is played by A, and solutions take the form e^(λt)v where λ and v are eigenvalue-eigenvector pairs of A. The matrix form makes this analogy explicit and systematic.