Consider solving an initial value problemy′(t) = f (t, y), y(0) = y0.(i) Derive the equations of the general step of the generalised Heun’s method, whichevaluates the right–hand side f (t, y) at the points t = tn and t = tn + 14 h in eachinterval [tn, tn + h].

Apply the generalised Heun’s method from part (a)(i) to solve the initial value prob-lemy′(t) = 11 + y2 , y(0) = 0.Use 2 steps of the algorithm with the step size h = 1 to approximately find y(2).Show all calculations with at least 6 accurate decimal digits

Consider the initial value problemy′(t) = cos(y), y(0) = 0.(i) Apply 4 steps of forward Euler method with the step size h = 12 to approximately findy(2). Show at least 4 decimal places in your calculations. [10 marks](ii) Apply 2 steps of Heun’s method with the step size h = 1 to approximately find y(2).Show at least 4 decimal places in your calculations.

Consider the following population dynamics model, proposed by Pierre–FrançoisVerhulst in 1838: dydt = ry(t)(1 − y(t)K), y(0) = y0.Here, y(t) denotes the population size at time t, y0 denotes the initial population size, r denotes the growthrate, and K denotes the carrying capacity of the environment.The analytic solution for this equation isy(t) = Ky0ertK + y0(ert − 1).(a) Implement the generalised Heun’s algorithm with α = 12 (also known as the modified Euler’s method) forthis problem.Apply this algorithm to numerically solve the equation for t ∈ [0, T ], using the uniform grid t0, t1, . . . , tNwith t0 = 0, tN = T.Your code should be flexible and able to work for general r, y0, K, T and N.Your code should not use functions from Matlab Symbolic package, such as syms , diff and dsolve .Implement your program in Matlab as a single file and save it with the name q2_2106015.m

The analytic solution for the initial value problem from part (a)(ii) isy(t) =(32 t +√1 + ( 32 t)2)1/3−(32 t +√1 + ( 32 t)2)−1/3.Calculate the accuracy of the approximation obtained by the generalised Heun’s meth-ods at point t = 2.

principle of turam method