Project 1This project introduces approximations to Ordinary Diferential Equations using numerical methods. You will program three numerical solvers: Eulers method, Improved Eulers method, and 4thorder Runge-Kutta (RK4). You are required to write your own numerical methods in either MATLAB or MS Excel. You are not allowed to use numerical solvers written by anyone else.Problem 1: Consider the Following Initial Value Problem (IVP) where yis the dependent variable and tis the independent variable:y'=sin(t)(1y)with y(0)=y0t 0Note: the analytic solution For this IVP is:y(t)=1+(y01)ecos(t)1Part 1A: Approximate the solution to the IVP using Eulers method with the Followingconditions: Initial condition y0=12; time step h=116; and time intervalt[0,20]+ Derive the recursive Formula For Eulers method applied to this IVP+ Plot the Eulers method approximation+ Plot the absolute error between the approximation and the exact solution using a semilog plotPart 1B: Approximate the solution to the IVP using the Improved Eulers method with the Following conditions: Initial condition y0=12; time step h=116; and time interval t[0,20]+ Derive the recursive Formula For the Improved Eulers method applied to this IVP+ Plot the Improved Eulers method approximation+ Plot the absolute error between the approximation and the exact solution using a semilog plotPart 1C: Approximate the solution to the IVP using the RK4 method with the Following conditions: Initial condition y0=12; time step h=116; and time interval t[0,20]+ Plot the RK4 method approximation+ Plot the absolute error between the approximation and the exact solution using a semilog plot Background image of page 1 Project 1

Problem 2: Consider the following Initial Value Problem (IVP) where y(t)is the dependent function:y'=yy2+1.14 cos(et/2)with y(0) =y0t 0Part 2A: Approximate the solution to the IVP using the Improved Eulers method with the following conditions: Initial condition y0=1; time stepsh=18,116,132,164; and time interval t[0,20]+ Plot the Improved Eulers method approximation for all 4 time steps+ Discuss the results of these approximationsPart 2B: Approximate the solution to the IVP using the RK4 method with the following conditions: Initial condition y0=1; time steps h=1/8,1/16,1/32,1/64; and time interval t[0,20]+ Plot the RK4 approximation for all 4 time steps+ Discuss the results of these approximations