Using Runge-Kutta method of order 4 to approximate y(1) with
step size h = 0.1 and h = 0.2 respectively (keep 8 decimals):
dy/dx = x + arctan y, y(0) = 0.
Solutions: when h = 0.1, y(1) = 0.70398191. when h = 0.2, y(1) =
0.70394257.
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]
Prompt: Produce a 4th order Runge Kutta code in
PYTHON that evaluates the following second order
ode with the given initial conditions, (d^2y/dt^2) +4(dy/dt)+2y=0,
y(0)=1 and y'(0)=3. After your code can evaluate the 2nd order ode,
add a final command to plot your results. You may only
use python.
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
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 3 steps of the Runge-Kutta (fourth order) method to solve
the following differential equation to t = 2.4, given that y(0) =
2.3. In your working section, you must provide full working for the
first step. To make calculations easier, round the tabulated value
of y at each step to four decimal places.
a) Provide the four K-values that are calculated at the first step,
with four decimal places. b) Provide your answer for y(2.4) with
four decimal places....
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.
With the aid of fourth order Runge-Kutta method, solve the
competing species model defined by
dx/dt =x(2 − 0.4x − 0.3y), x(0) = 2
dy/dt =y(1 − 0.1y − 0.3x), y(0) = 4
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.