Question: Methods for Higher Order ODEs and for Systems of Simultaneous ODEs

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.

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