Question

Q3p please with mathlab.

1) The growth of populations of organisms has many engineering and scientific applications. One of the simplest models assumes that the rate of change of the population p is proportional to the existing population at any time t: dp/dt = kp where k, is the growth rate. The world population in millions from 1950 through 2000 was 1950 1959 1960 1965 1970 1975 1980 1989 1990 19952000 P25602780 3040 3350 3710 4090 4450 4850 528056906080 . Use the data from 1950 to 1970 to estimate k, by using a numerical method. b) Use the fourth-order Runge-Kutta method to determine the world population in 2005 and 2010. c) Determine the world population in 2015 by using Adam's fourth-order method. 2) Solve the following problem over the interval from x=0 to 0.5 using a step size of 0.25 where y(0=1. dy/dx = (1+4x)y1/2 a) Analytically, b) using simple Euler, c) improved Euler and d) midpoint Euler methods. 3) Write a computer program that may be used to solve problem 2 by using midpoint Euler method. Here, take the step size h=0.1 to obtain the solution over the interval x=0 to 1. 4) The van der Pol equation is a model of an electronic circuit that arose back in the days of vacuum tubes. dy/dt? - (1-y) dy/dt + y = 0 Given the initial conditions, yO=y'(0=1, solve this equation from t=0 to 1 by using simple Euler method with a step size of 0.25.

Answer 1

clc;clear all
%%Midpoint-Euler method
dydx=@(x,y) (1+4.*x).*sqrt(y); %%Given ODE
a=0;b=1;h=0.1;
[xa ya]=MidEuler(dydx,a,b,h,1) %%Function call

function [x y]=MidEuler(ODE,a,b,h,yini) %%Function for Mid-Point Euler method
x(1)=a;y(1)=yini; %%Initial conditions
n=(b-a)/h; %%Intervals
for(i=1:n) %%Midpoint Euler method
xm=x(i)+0.5*h;
ym=y(i)+ODE(x(i),y(i))*0.5*h;
x(i+1)=x(i)+h;
y(i+1)=y(i)+ODE(xm,ym)*h;
end
end

1 2 3 - clc;clear all Midpoint-Euler method dydx=@ (x, y) (1+4.*x).*sqrt(y); Given ODE a=0; b=1;h=0.1; [xa ya] =MidEuler (dydx, a, b, h, 1) Function call 4 - 5 - 6 7 8 function [x y)=MidEuler (ODE, a, b, h, yini) Function for Mid-Point Euler method 9 x(1)=a; y (1)=yini; %%Initial conditions 10 n=(b-a) /h; %%Intervals 11 for(i=1:n) Midpoint Euler method 12 xm=x (i) +0.5*h; 13- ym=y (i) +ODE (x(i),y(i))*0.5*h; 14 x(i+1)=x (i) +h; 15 - y(i+1)=y(i) +ODE (xm, ym) *h; 16 - end 17 - endl Command Window xa = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000 ya = 1.0000 1.1230 1.2980 1.5347 1.8450 2.2431 2.7457 3.3721 4.1437 5.0845 6.2207