GATE AG 2008 • Q26 MCQ • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)

Question Statement:

The solution of the linear differential equation $\frac{dy}{dx} - 7y = e^x$ is:

Options:

  • (A) eˣ(Ce⁶ˣ − 6⁻¹)
  • (B) eˣ[e⁷ˣ − (6eˣ)⁻¹]
  • (C) eˣ(Ce⁵ˣ − 6⁻¹)
  • (D) eˣ[Ce⁵ˣ + (6eˣ)⁻¹]

Official Answer Key:

A

Step-by-Step Solution & Explanation

(A) eˣ(Ce⁶ˣ − 6⁻¹) (B) eˣ[e⁷ˣ−(6eˣ)⁻¹] (C) eˣ(Ce⁵ˣ − 6⁻¹) (D) eˣ[Ce⁵ˣ+(6eˣ)⁻¹] Integrating factor = e^(−7x). d/dx(y·e^(−7x)) = eˣ·e^(−7x) = e^(−6x). Integrate: y·e^(−7x) = −(1/6)e^(−6x) + C → y = Ce^(7x) − (1/6)eˣ = eˣ(Ce^(6x) − 1/6). Section 2 — Farm Machinery

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