Questions: Integrating Factor Method for First-Order Linear ODEs

Question 1

A student correctly computes μ(x) = x³ for the ODE y′ + (3/x)y = x² and multiplies through to get x³y′ + 3x²y = x⁵. They then integrate x³y′ and 3x²y separately to get (x⁴/4)y and (3x³/3)y. What error did they make?

• A. The integrating factor should be e^(−3/x), not x³
• B. The right side x⁵ should have been integrated before multiplying by μ
• C. After multiplying by the integrating factor, the left side x³y′ + 3x²y must be recognized as d/dx[x³y] and written as a single derivative before integrating — the two terms cannot be integrated separately
• D. The constant of integration was omitted from the integrating factor

Answer: (C)

The entire purpose of the integrating factor is to convert the two-term left side into a single derivative d/dx[μy] via the product rule. Once you verify x³y′ + 3x²y = d/dx[x³y], you must write the equation as d/dx[x³y] = x⁵ before integrating both sides to get x³y = x⁶/6 + C. Integrating the terms separately is wrong because each term is not individually a complete derivative — the product rule collapse is the whole mechanism that makes the method work.

Question 2

Why is the constant of integration dropped when computing the integrating factor μ(x) = e^(∫P(x)dx)?

• A. Constants of integration are never needed in differential equations
• B. Including the constant gives μ = e^C · e^(∫P dx); the extra factor e^C multiplies both sides of the equation and cancels, so any particular choice of the constant gives an equally valid integrating factor — the simplest (C = 0) is used
• C. The integrating factor must equal 1 at x = 0, which forces the constant to be zero
• D. Including the constant makes the resulting ODE nonlinear

Answer: (B)

Any nonzero multiple of μ works as an integrating factor. Including a constant of integration C gives μ_general = e^(∫P dx + C) = e^C · μ₀. Multiplying both sides of the ODE by e^C · μ₀ is the same as multiplying by μ₀ — the e^C cancels. Dropping the constant (taking C = 0) gives the simplest valid integrating factor, which is all we need. This is one of the few places in ODE solving where omitting the constant of integration is not an error.

Question 3

The integrating factor method can solve any first-order ODE, including nonlinear equations like y′ = y².

• T. True
• F. False

Answer: False

The integrating factor method is specifically designed for first-order *linear* ODEs of the form y′ + P(x)y = Q(x). It works by exploiting the algebraic structure of linear equations: the left side can be made into a product-rule derivative after multiplication by a carefully chosen function. Nonlinear equations like y′ = y² lack this structure — you cannot convert their left side into d/dx[μy] for any μ. Separation of variables, substitution methods, or other techniques are needed for nonlinear equations.

Question 4

After multiplying a first-order linear ODE by its integrating factor, the left side becomes the derivative of a product and can be directly integrated without further manipulation.

• T. True
• F. False

Answer: True

This is the central mechanism of the method. The integrating factor μ(x) = e^(∫P dx) is derived precisely so that μ′ = μP. After multiplying y′ + P(x)y = Q(x) by μ, the left side becomes μy′ + μPy = μy′ + μ′y = d/dx[μy]. The equation is now d/dx[μy] = μQ, which integrates directly to μy = ∫μQ dx. The two-term left side collapses into a single derivative — this collapse is what makes the ODE solvable.

Question 5

Explain why the integrating factor μ(x) = e^(∫P(x)dx) transforms the ODE y′ + P(x)y = Q(x) into a directly integrable form. Where does the formula for μ come from?

Model answer: The goal is to find a function μ(x) such that μ(y′ + Py) equals d/dx[μy]. Expanding by the product rule: d/dx[μy] = μy′ + μ′y. For this to equal μy′ + μPy, we need μ′ = μP. This is a separable ODE for μ: dμ/μ = P dx, which integrates to ln|μ| = ∫P dx, giving μ = e^(∫P dx). With this choice, multiplying the ODE by μ makes the left side exactly d/dx[μy] = μQ, which integrates to μy = ∫μQ dx.

The integrating factor is not a formula to be memorized and applied mechanically — it is derived by asking 'what function would make the left side a product-rule derivative?' This question has a unique answer (up to a constant multiple), and that answer is e^(∫P dx). Understanding the derivation means you can reconstruct the formula if you forget it, adapt the method to related problems, and explain why it works to someone else — all hallmarks of genuine understanding rather than procedural fluency.