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Solving systems of differential equations In the last lecture, we discussed the following question: For some interval I C R, find a function g: I - R such that dy for some known function f I X y(I) R. We also retalised that we cannot solve all differential equations on paper. We can also consider sysfems of differential equations, such as the following initial value problems of two differential equations. For some interval [a, 5] C R, find functions 01.82 [a,b] - R such that vi(x) = (1) U2(x) for some known functions fi and f2. together with the initial conditions gi(a) = d1 and g2(a) = d2. For instance, we can have the following example. Example 1 Let a = 0, 5 = 1 and consider the system with gn(0) = 1 and g2(0) = 0. Upon defining the vector function F: R³ -IR with = and y = we can write the initial value problem (1) as y'(x)=F(x,y) = y(a) =(d1,d2² = (2) with the notation y' (x) meaning Problem 1 We would like to approzimate the solution to the above system. De-rive a generalisation of the foraward Euler method for one equation to this case of a sgstem of fao equations. Problem 2 Wrife an algorithm for the foratard Euler method you derived for the system above. Problem 3 Write a MATLAB" function on paper implementing the vector func-tion F for Example 1. Problem 4 During the class, write a MATLABR programme on paper, implementing the foraward Euler method you derived above for the initial value problem of Ezample 1. After the class, import the MATLABR programme you wrofe, correct it (if there are any mistakes) and run it for 10, 20, 50, 100 subinteroals. Plot the graphs of the approximations of y an 1/2 against 2, for 100 subinterzals on the same plof. Exfend the interval of Ezample 1 from [O, 1] to [0, 5]. Run the programme for a suitable number of subinfervols to achieve similar accuracy to the plot above.

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