GATE AG 2007 • Q4 MCQ • 1 Mark

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

Taking $y(0) = 0$ and using Euler's method with step size $h = 0.1$, the solution of the differential equation $\frac{dy}{dx} = x + y$ at $x = 0.2$ is:

Options:

  • (A) 0.3101
  • (B) 0.3142
  • (C) 0.6202
  • (D) 4.080

Official Answer Key:

A

Step-by-Step Solution & Explanation

Euler's method: y_{n+1} = y_n + h·f(x_n,y_n), f(x,y)=2xy+1. Step1: x=0,y=0 → y=0+0.1(1)=0.1000. Step2: x=0.1,y=0.1 → f=2(0.1)(0.1)+1=1.02 → y=0.1+0.1(1.02)=0.2020. Step3: x=0.2,y=0.2020 → f=2(0.2)(0.2020)+1=1.0808 → y=0.2020+0.1(1.0808)=0.3101. (B) is the exact-ish value from a finer method; (C),(D) come from arithmetic/step errors.

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