Question

Question 9 10 pts Given the ODE, initial condition, and step size: dx + 2x3 = e-3t x(3) = 1 and At = 0.5. We intend to find x(t=3.5) using the Runge Kutta 4 method. Find all the slopes and the required increment function for this step. Input your answers to 4 decimal places increment function =

Answer 1

By Runge Kutta Method,

y2 = y1 + 3 (k1 + 2.62 +2.k3+k4)

kl = hf 21, yl)

k2 = hf (01 + y +

k3 = hf (21+ y +

k4 = hf(x1+h, yl + k3)

Here,

f(t, x) = -34 -2.0-3

r1 = x(3) = 1, h = 0.5

c2 = (3.5)

k1 = hf (t1, x1) = 0.5f(3, 1) = -0.9999

0.5 k2 = hf(t1 + 21+ ) = 0 f(3 f(3+ 1+- -0.9999 -) = -7.9999

J:3 = hs(t1 + 5, r1 + -7.9999、 ー) = 0.0371 13.05 tp1+

k4 = hf (t1 +h, rl + 3) = 0.5f(3+0.5,1 +0.0371) = -0.896

Thus,

-0.9999 + 2(–7.999) + r2 = rl+ =(k1 + 2.k2 + 2.k3 + k4) = 1 2(0.0371) - 0.896)

=1+(-2.97025) = -1.97025

Increment is (-2.97025).

Increment function is,

0 = =(x1 +2.2 + 2.83 + 4)