## 15 Reputation

1 years, 231 days

## Methods for Higher Order ODEs and for Sy...

Maple
``````my"+cy'+ky=F(t)      y(0)=y_0    and y'(0) =y'_0    (1)
``````

Where F(t) is an external force, m, c and k are positive constants, y is a function of t and cy' is the damping term. Use equation (1) for your topic and F(t) = 0.

Task 2 : Assume that m = 1 unit in equation (1). For each of the following cases (all with m = 1), perform 5 steps of the modified Euler method applied to the coupled first order system you obtained in part (i), taking the step size h = 0.02 and working to 5 decimal places at each stage :

``````c = 0,  k = 4  , y(0)=1   and y'(0) =0;

c = 2,  k = 0,   y(0)=1   and y'(0) =0;

c = 2,  k = 1,   y(0)=1   and y'(0) =0;

c = 4, k = 6.25,  y(0)=1   and y'(0) =0.
``````

Summarise your step-by-step results for each example in a table like the blank template below.

 c = (Value),  k = (Value), h = 0.02, t (Predicted) (Corrected) (Predicted) (Corrected) 0.00 1.00000 1.00000 0.00000 0.00000 0.02 0.04 0.06 0.08 0.10

This is a question based on rewriting the equation as a pair of coupled(simultaneous) first order ODEs of the general form.

If anyone could please help to proceed with this question, as I am not quiet sure how to approach this differential equation. I will really appreciate your help please.

Thank you

## Direction field using Maple...

Maple

Suppose that a given population can be divided into two parts, those who have a given disease and can infect others, and those who do not have it but susceptible. Assume that y the proportion of infectious individuals then the rate of spread dy/dt is proportional to the number of contacts and can be described as

=y(-y), y(0)=y0

Where y > 0 is a function of t,  is the initial proportion of infectious individuals.

1. Use MAPLE and sketch a direction field for your differential equation and include a sufficient number of solution curves and include the graph into your answer sheet.

(3 marks)

1. Find all the equilibrium solutions and determine whether they are asymptotically stable or unstable.

(4 marks)

1. Solve the above initial value problem and verify that the conclusions you reached in part(ii) are correct.

(5 marks)

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