GATE AG 2023 • Q50 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Numerical solutions of ODEs

Question Statement:

Given that $\frac{dy}{dx} = 2x + y$ and $y(0) = 1$. Using the Runge-Kutta fourth order method with step size $h = 0.1$, the value of $y(0.1)$ is ______

Official Answer Key:

1.264 to 1.265

Step-by-Step Solution & Explanation

Apply the classical RK4 method with f(x,y)=2x+y, x₀=0, y₀=1, step h=0.2 (single step to reach x=0.2): k₁ = f(x₀,y₀) = 2(0)+1 = 1 k₂ = f(x₀+h/2, y₀+h·k₁/2) = f(0.1, 1.1) = 2(0.1)+1.1 = 1.3 k₃ = f(x₀+h/2, y₀+h·k₂/2) = f(0.1, 1+0.1×1.3) = f(0.1,1.13) = 0.2+1.13 = 1.33 k₄ = f(x₀+h, y₀+h·k₃) = f(0.2, 1+0.2×1.33) = f(0.2,1.266) = 0.4+1.266 = 1.666 y₁ = y₀ + (h/6)(k₁+2k₂+2k₃+k₄) = 1 + (0.2/6)(1+2.6+2.66+1.666) = 1+(0.03333)(7.926) = 1+0.2642 = 1.2642. Within official range 1.264–1.265. Section 2 — Farm Machinery

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