Question

Consider the differential equation below dx/dt = −x with an initial condition x(t=0)=1. Write a program that computes x(t) using second-order Runge-Kutta method. Find the relative error of the computed value as compare to the exact value after 20 steps for τ=0.2

Answer 1

To compute x(t) using the second-order Runge-Kutta method for the given differential equation dx/dt = -x with the initial condition x(t=0) = 1, follow the steps mentioned in the detailed answer.

Step-by-step explanation:

To solve the given differential equation dx/dt = -x with the initial condition x(t=0) = 1 using the second-order Runge-Kutta method, you can follow these steps: Set the time step size, τ, to 0.2. Initialize the variables x and t to their initial values: x = 1 and t = 0. Repeat the following steps 20 times: Calculate the exact value of x after 20 steps. Calculate the relative error between the computed value and the exact value.

The student is asking for a program that computes the solution to the differential equation dx/dt = −x using the second-order Runge-Kutta method (RK2), given an initial condition x(t=0) = 1. After computing the solution after 20 steps with a step size of 0.2, they also want to find the relative error compared to the exact solution. The exact solution to this differential equation is x(t) = e−t, which we can use to calculate the relative error. To calculate the relative error, you can use the following formula: Relative Error = (Exact Value - Approximate Value) / Exact Value. By finding the values of x(t) at each step using the RK2 algorithm and then comparing them to the exact values, you can determine the relative error after 20 steps.