In: Math

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.

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.

Write a user-defined MATLAB function that uses classical fourth
order Runge-Kutta method to solve a first order ODE problem dydx =
f(x, y) in a given interval a ? x ? b with initial condition y(a) =
y0 and step size of h. For function name and arguments, use [x,y] =
myrk4(f, a, b, h, y0)
Check your function to find the numerical solution for
dydx=?1.2y+7e^(?0.3x) in the interval 0 ? x ? 4 with initial
condition y(0)=3. Run your...

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....

1)Select all that applies to the Fourth-order Runge-Kutta (RK4)
method K subscript
1 equals f left parenthesis t subscript k comma y subscript k
right parenthesis K subscript
2 equals f left parenthesis t subscript k plus h over 2 comma
space y subscript k plus h over 2 space K subscript 1 right
parenthesis K subscript
3 equals f left parenthesis t subscript k plus h over 2 comma
space y subscript k plus h over 2 space K...

Use Classic Runge-Kutta method with h = 1 to solve the
system y” - y’ - 6y = 0, y(0) = 2, y’(0) = 3 on [0,1]

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)

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.

Using Runge-Kutta method, compute y(0.3), from the equation dy
dx = xy 1+x2 with y(0) = 1, take h = 0.1

Solve the system of equation by method of elimination.
dx/dt + x−5y = 0,
4x +dy/dt+ 5y = 0,
x(0) = −1, y(0) = 2.

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.

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