IGNOU MPH 14 2026 - Derive the Runge-Kutta 2nd Order formula for the differential equation

Question

Derive the Runge-Kutta 2nd Order formula for the differential equation

$$\frac{dy}{dx} + y = \exp(-x)$
Use this to determine the value of y at $x = 2$ given that $y(x = 0) = 2$ , using a step size $h = 1.0$ .

19 Jan 2026

Answer :

Word Count : 303

We are asked to solve the differential equation: [ \frac{dy}{dx} + y = e^{-x}, \quad y(0) = 2 ] using the 2nd-order Runge-Kutta method (RK2) with step size (h = 1.0) up to (x = 2). --- ### Step 1: Standard RK2 formula For a differential equation (\frac{dy}{dx} = f(x, y)), the RK2 method is: [ k_1 = h f(x_n, y_n) ] [ k_2 = h f(x_n + h, y_n + k_1) ] [ y_{n+1} = y_n + \frac{1}{2} (k_1 + k_2) ] Here, (f(x, y) = e^{-x} - y) (rearranging (\frac{dy}{dx} = e^{-x} - y)). --- ### Step 2: Initialize values [ x_0 = 0, \quad y_0 = 2, \quad h = 1 ]

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