Question

MATLAB HELP 3. Consider the equation y′ = y2 − 3x, where y(0) = 1.   USE THE EULER AND RUNGE-KUTTA APPROXIMATION SCRIPTS PROVIDED IN THE PICTURES

a. Use a Euler approximation with a step size of 0.25 to approximate y(2).

b. Use a Runge-Kutta approximation with a step size of 0.25 to approximate y(2).

c. Graph both approximation functions in the same window as a slope field for the differential equation.

d. Find a formula for the actual solution (not by hand). Which is most helpful in understanding the solution: the explicit formula, or the numerical approximation? Explain.

e. Use the explicit formula to compute the exact value of y(2). How close are the Euler and Runge-Kutta approximations?

Please help me form the MATLAB code needed for this question. USE THE EULER AND RUNGE-KUTTA APPROXIMATION SCRIPTS PROVIDED

3. Consider the equation y' =y®-3x, where y(0) = 1. a. Use an Euler approximation with a step size of 0.25 to approximate y(2) b. Use a Runge-Kutta approximation with a step size of 0.25 to approximate y(2). c. Graph both approximation functions in the same window as a slope field for the differential equation. d. Find a formula for the actual solution (not by hand). Which is most helpful in understanding the solution: the explicit formula, or the numerical approximation? Explain. e. Use the explicit formula to compute the exact value of y(2). How close are the Euler and Runge-Kutta approximations? Euler approximation function [x,y] = EulerApprox(dydx , xrange,yo,h) xxrange (1):h:xrange (2) x transpose(x): y-zeros (size ( while(i

Answer 1

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AS FOR GIVEN DATA.

%%Matlab code for Euler's forward and Rk4 method
clear all
close all
%Answering question a.
%---------------------%
%Program for Euler
%function for Euler equation solution
syms x y
f(x,y)=y^2-3*x;
%Euler steps for h and x_end
%Initial values
x0=0;
y0=1;
%step size
h=0.25;
%x end value
xend=2;
xn=x0:h:xend;
% Euler steps
y_result1(1)=y0;
x_result1(1)=x0;
%Loop for Euler Steps
for i=1:length(xn)-1
    x_result1(i+1)= x_result1(i)+h;
    y_result1(i+1)=y_result1(i)+h*double(f(x_result1(i),y_result1(i)));
end
%printing the result for Euler
fprintf('\tThe approximate solution using Euler method at x=%d for h=%0.2f is %f\n',xend,h,y_result1(end))

%Answering question b.
%---------------------%
%Program for RK4
%function for RK4 equation solution
syms x y
f(x,y)=y^2-3*x;
%RK4 steps for h and x_end
%Initial values
x0=0;
y0=1;
%x end value
xend=2;
xn=x0:h:xend;
% RK4 steps
y_result2(1)=y0;
x_result2(1)=x0;

    %Runge Kutta 4 iterations
    n=(xend-x0)/h;
    for i=1:n
     
        k0=h*f(x_result2(i),y_result2(i));
        k1=h*f(x_result2(i)+(1/2)*h,y_result2(i)+(1/2)*k0);
        k2=h*f(x_result2(i)+(1/2)*h,y_result2(i)+(1/2)*k1);
        k3=h*f(x_result2(i)+h,y_result2(i)+k2);
        x_result2(i+1)=x_result2(i)+h;
        y_result2(i+1)=double(y_result2(i)+(1/6)*(k0+2*k1+2*k2+k3));
      
    end
%printing the result for RK4
fprintf('\tThe approximate solution using RK4 method at x=%d for h=%0.2f is %f\n',xend,h,y_result2(end))

%Answering question c.
%---------------------%
hold on
%plotting the solutions for Euler and RK4
plot(x_result1,y_result1,'b')
plot(x_result2,y_result2,'g')
title('Solution plot for differential equation')
xlabel('x')
ylabel('y')
%slope field calculation
funn=@(x,y) y.^2-3.*x;
xx=0:.2:2; yy=-3:.2:3;
[X,Y]=meshgrid(xx,yy);
slopes=funn(X,Y);
dy=slopes./sqrt(1+slopes.^2);
dx=ones(size(dy))./sqrt(1+slopes.^2);
h=quiver(X,Y,dx,dy,.5,'color','r');
set(h,'maxheadsize',0.1)
legend('Euler solution','RK4 solution','Slope field','location','best')

%Answering question d.
%---------------------%
%Finding exact solution using dsolve
syms y(x)
eqn=diff(y,x)==y^2-3*x;
cond=y(0)==1;
y_result3(x)=dsolve(eqn,cond);
%printing exact solution
fprintf('\tThe exact solution using dsolve at x=%d is %f\n',xend,y_result3(xend))
fprintf('The expression for Exact solution\n')
disp(y_result3(x))
fprintf('Numerical solution are better to understand than exact solution.\n')
%Answering question e.
%---------------------%
err1=double((abs(y_result3(xend)-y_result1(end))/y_result3(xend))*100);
err2=double((abs(y_result3(xend)-y_result2(end))/y_result3(xend))*100);
fprintf('For step size h=%2.2f.\n',h)
fprintf('\tThe percentage relative error for solving differential eqn using Euler method is %f\n',abs(err1))
fprintf('\tThe percentage relative error for solving differential eqn using RK4 method is %f\n',abs(err2))

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