Questions: First-Order Linear Ordinary Differential Equations

Question 1

The equation dy/dx + 3y = 6 is solved using an integrating factor. As x → ∞, what determines the long-run behavior of the solution?

• A. The constant of integration C, which scales the entire solution
• B. The homogeneous solution Ce^(−3x), which stabilizes to a nonzero value
• C. The particular solution y = 2, which the system settles toward as the transient decays
• D. The forcing term Q(x) = 6, which continues to drive the system indefinitely

Answer: (C)

The general solution is y = 2 + Ce^(−3x). As x → ∞, the homogeneous part Ce^(−3x) → 0 (the transient dies away), leaving only y = 2 — the particular solution and the system's steady state. The forcing term Q(x) = 6 is what *creates* the steady state, but the steady state itself is the particular solution y_p = 2, not the forcing term directly.

Question 2

Which property of the equation dy/dx + P(x)y = Q(x) is what makes the integrating factor method work and guarantees the solution has a homogeneous-plus-particular structure?

• A. Q(x) must be a continuous function
• B. P(x) must be a positive constant
• C. Both y and dy/dx appear only to the first power — the equation is linear in y
• D. The equation must have a unique solution for every initial condition

Answer: (C)

Linearity — y and dy/dx appearing only to the first power, with no y², sin(y), or (dy/dx)³ terms — is the structural property that makes the integrating factor trick work. It guarantees that the solution space has the additive structure that allows homogeneous and particular solutions to be combined. Nonlinear first-order ODEs do not generally admit this clean decomposition and cannot be solved by integrating factors.

Question 3

For the equation dy/dx + P(x)y = Q(x) with P(x) > 0, the homogeneous solution always decays to zero as x → ∞.

• T. True
• F. False

Answer: True

The homogeneous solution is y_h = Ce^(−∫P(x)dx). When P(x) > 0, the integral ∫P(x)dx grows without bound as x → ∞, so e^(−∫P(x)dx) → 0. This is why the homogeneous part is called a 'transient' in applied contexts — it represents the system's natural response to initial conditions, which eventually dies away, leaving only the particular solution (the forced steady state).

Question 4

In the general solution y = y_h + y_p of a first-order linear ODE, y_h captures the steady-state behavior the system approaches, while y_p represents the transient that dies away over time.

• T. True
• F. False

Answer: False

This is reversed. The homogeneous solution y_h is the transient — it decays to zero (when P(x) > 0) and represents what the system does when left to its own natural dynamics from initial conditions. The particular solution y_p is the steady state — the system's persistent response to the forcing term Q(x). In the example dy/dx + 2y = 4, the solution y = 2 + Ce^(−2x) has transient Ce^(−2x) (homogeneous) and steady state y = 2 (particular).

Question 5

Explain why the solution to a first-order linear ODE dy/dx + P(x)y = Q(x) splits into a homogeneous part and a particular solution. What does each part represent physically or dynamically?

Model answer: The homogeneous part y_h solves the equation with Q(x) = 0 — it describes the system's natural behavior driven purely by initial conditions, with no external forcing. The particular solution y_p represents the system's response to the forcing term Q(x). The full solution combines both: y_h captures how the initial state evolves (usually decaying), while y_p captures where the system is being driven toward. Linearity of the equation is what guarantees these two parts simply add together.

This structure — natural response plus forced response — appears throughout applied mathematics because linear systems obey superposition. In physical terms: the transient (y_h) is what the system 'wants' to do based on where it started; the steady state (y_p) is what the external driver is pushing it toward. As time passes, the transient fades and the forced behavior dominates. This is why circuits eventually charge to supply voltage, why temperatures equilibrate, and why chemical concentrations approach equilibrium.