Question

Find approximate value of y(0,1) for initial value problem
y'=x^3-y^3 , y(0)=1 using multiple method

\begin{equation} y_{n+1} = y_n + \frac{h}{3}(7f_n - 2f_{n-1} + f_{n-2}) \end{equation}

with h=0.2. Calculate the starting values using Runge-Kutta second order method with the same h.

15 Feb 2024
Answer :
Word Count : 320

First, let's start with the Runge-Kutta second order method to get the starting values:

Given the initial value problem:

\[
\frac{{dy}}{{dx}} = x^3 - y^3 \quad \text{with} \quad y(0) = 1
\]

Using the second-order Runge-Kutta method:

\[
k_1 = h \cdot f(x_n, y_n)
\]
\[
k_2 = h \cdot f(x_n + \frac{h}{2}, y_n + \frac{k_1}{2})
\]

where \(f(x, y) = x^3 - y^3\).

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