In: Advanced Math

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.

(dy /dt) = 1.1ty

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 4 steps of the Modiﬁed Euler’s method to solve the following
diﬀerential equation to t = 2.6, given that y(0) = 1.1. In your
working section, you must provide full working for the ﬁrst two
steps. To make calculations easier, round the calculations at each
step to four decimal places, and provide your ﬁnal answer with four
decimal places. dy/ dt = 1.4sin(ty)

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)

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 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 method of undetermined coefficients (it works the same
way for a fourth order linear equation) to find the general
solution of:
y(4) - 18y'' + 81y = 6t2 - 4t + 36 +
90e3t
[Hint: The complementary solution is yc =
C1e3t + C2e-3t +
C3te3t + C4te-3t

Use the simplex method to solve the following linear programming
problems. Clearly indicate all the steps, the
entering and departing rows and columns and rows, the pivot
and the row operations used. An investor has up to
N$450,000 to invest in three types of investments. Type A pays 6%
annually and has a risk factor of 0. Type B pays 10% annually and
has a risk factor of 0.06. Type C pays 12% annually and has a risk
factor of...

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