Given 3
/
+ ^
2 = ^
, y(0.3) = 5 and using a step size
of h=0.3, the value of y(0.9) using Runge-Kutta 4th order method is
most nearly
(i)-1.6604 (ii) – 1.1785 (iii) - 0. 45831 (iv) 2.7270
Answers
Answer:
Runge-Kutta 4th Order Method to Solve Differential Equation
Given following inputs,
An ordinary differential equation that defines value of dy/dx in the form x and y.
Initial value of y, i.e., y(0)
Thus we are given below.
The task is to find value of unknown function y at a given point x.
The Runge-Kutta method finds approximate value of y for a given x. Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method.
Below is the formula used to compute next value yn+1 from previous value yn. The value of n are 0, 1, 2, 3, ….(x – x0)/h. Here h is step height and xn+1 = x0 + h
. Lower step size means more accuracy.
The formula basically computes next value yn+1 using current yn plus weighted average of four increments.
k1 is the increment based on the slope at the beginning of the interval, using y
k2 is the increment based on the slope at the midpoint of the interval, using y + hk1/2.
k3 is again the increment based on the slope at the midpoint, using using y + hk2/2.
k4 is the increment based on the slope at the end of the interval, using y + hk3.
The method is a fourth-order method, meaning that the local truncation error is on the order of O(h5), while the total accumulated error is order O(h4).