Use the Runge-Kutta method with step sizes h = 0.1, to find
approximate values of the solution of
y' + (1/x)y = (7/x^2) + 3 , y(1) = 3/2 at x = 0.5 .
And compare it to thee approximate value of y = (7lnx)/x +
3x/2
Use the Runge-Kutta method and the Runge-Kutta semilinear method
with the indicated step sizes to find approximate values of the
solution of the given initial value problem at 11 equally spaced
points (including the endpoints) in the interval. This question is
from the differential equation.
y'-4y = x/y^2(y+1) , y(0) = 1; h=0.1, 0.05 , 0.025, on [0,
1]
1. Write a python code that uses the Runge Kutta Method method
to approximate the solutions to each of the following initial-value
problems and compare/plot the results to the actual values.
a) y′=te^(3t) − 2y, 0 < t < 1, y(0) = 0
with h = 0.5; actual solution y(t)=1/5te^(3t) − 1/25e^(3t) +
1/25e^(−2t).
- Use the Runge Kutta method to approximate/plot the solutions
to each of the following initial-value
b) ?′=1+(?−?)2,2<?<3,?(2)=1y′=1+(t−y)2,2
c) ?′=1+??,1<?<1,?(1)=2y′=1+yt,1
Problem Four
Use Runge Kutta method of order four to approximate the solution
of the initial value problem
?′ + 2? = ??3?, 0 ≤ ? ≤ 1, ?(0) = 0, ???ℎ ℎ = 0.5
Hint: Compute ?(0.5) ??? ?(1)
use Runge Kutta 4th order method
y'=y-1.3333*exp(0.6x)
a) h=2.5 and compare the value to the exact value
b) h=1.25 and compare the value to the exact value
Thks!
Q 4. With the aid of fourth order Runge-Kutta method, solve
the competing species model
[20 points]
defined by
dx =x(2 − 0.4x − 0.3y), x(0) = 4 dt
dy =y(1 − 0.1y − 0.3x), y(0) = 3 dt
where the populations x(t) and y(t) are measured in thousands
and t in years. Use a step size of 0.2 for 0 ≤ t ≤ 2 and plot the
trajectories of the populations with Matlab or GNU Octave.
Use Euler's Method with step size 0.11 to approximate y (0.55)
for the solution of the initial value problem
y ′ = x − y, and y (0)= 1.2
What is y (0.55)? (Keep four decimal places.)