Question

The differential equation : dy/dx = 2x -3y , has the initial conditions that y = 2 , at x = 0

Obtain a numerical solution for the differential equation, correct to 6 decimal place , using ,

1. The Euler-Cauchy method
2. The Runge-Kutta method in the range x = 0 (0.2) 1.0

Answer 1

Matlab code for Euler, Improved Euler and RK4 method clear all close all %Function for which solution have to do f-e (x,y) 2*x-3*y: fprintf( "\n\tEuler Method for step size h-%.2f\n',0.2) %Euler method h-0.2; x-0 Y-2; x_eval-1i n=(x-eval-x)/h; X2 ( 1 )#x; % amount of intervals s initial y % initial x at what point we have to evaluate Number of steps fprint f ( . At x(8.2f) for i-1: n solution of y is %f\n', x,y) 8Bular Steps m-double(f (x,y)); x-x+h y-y+h*m; x2(i+1)-xi fprint f (. At x(8.2E) solution of y is %f\n" , x,y) end fprintfnltThe solution using Euler Method %fin',h, x2 ( end ) , y2 ( end ) ) for h-8.2f at x(8.1f) is %RK4 method fprintf( "\n\tRK4 x= 0 ; Y-2; x eval-1 n«(x-eval-X)/h; Method for step size % initial x % initial y % at what point we have to evaluate % Number of steps h-%.2f\n',0.2) fprint f ( . At x(8.2t) for i-1: n RK4 Steps solution of y is %f\n', x,y) k1=h*double ( f ( x, y ) ) ; k2-h double (f (x+h/2), (ytkl/2); k 3-h double (f ((x+h/2), (y+k2/2))); k4-h*double (f ( (x+h), (ytk3))); dx (1/6) (k1+2*k2+2*k3+k4); y-y+dx; fprintf(" At x(8.2f) solution of y is %f\n',x,y)

end fprintf(ntThe solution using Runge Kutta 4 for h-s.2f at x(8.If) is 钍\n",h,x4(end ) , y4 ( end ) ) Plotting solution using Euler method figure(1) hold on plot (x2,y2, Linewidth',2) plot(x4,y4, Linewidth,2) xlabel ( 'x' ) ylabel'y(x)') title( Solution plot y vs.x' legend( Euler Method', 'RK4 Method', 'Location, 'northeast') grid on 응응88888888 End of Code 88888888888888 Euler Method for step size h=0.20 At x(0.00) solution of y is 2.000000 At x(0.20) solution of y is 0.800000 At x(0.40) solution of y is 0.400000 At x(0.60) solution of y is 0.320000 At x(0. 80) solution of y is 0.368000 At x(1.00 solution of y is 0.467200 The solution using Euler Method for h=0.20 at x(1.0) is 0.467200 RK4 Method for step size h-0.20 At x(0.00 solution of y is 2.000000 At x(0.20) solution of y is 1.132000 At x(0.40) solution of y is 0.715201 At x(0.60 solution of y is 0.546291 At x(0.80) solution of y is 0.51357.2 At x(1.00 solution of y is 0.555677 The solution using Runge Kutta 4 for h-0.20 at x(1.0) is 0.555677

Solution ploty vs. Euler Method RK4 Method 1.8 1.6 14 1.2 0.8 0.6 0.4 0.2 .1 0 03 04 05 0608 0.9 Published with MATLAB R2018a

%%Matlab code for Euler and RK4 method
clear all
close all
%Function for which solution have to do
f=@(x,y) 2*x-3*y;
fprintf('\n\tEuler Method for step size h=%.2f\n',0.2)

%Euler method
    h=0.2;           % amount of intervals
    x=0;             % initial x
    y=2;             % initial y
    x_eval=1;        % at what point we have to evaluate
    n=(x_eval-x)/h; % Number of steps
    x2(1)=x;
    y2(1)=y;
    fprintf(' At x(%.2f) solution of y is %f\n',x,y)
    for i=1:n
        %Eular Steps
        m=double(f(x,y));
        x=x+h;
        y=y+h*m;
        x2(i+1)=x;
        y2(i+1)=y;
        fprintf(' At x(%.2f) solution of y is %f\n',x,y)
    end
  
    fprintf('\n\tThe solution using Euler Method for h=%.2f at x(%.1f) is %f\n',h,x2(end),y2(end))

  
    %RK4 method
    fprintf('\n\tRK4 Method for step size h=%.2f\n',0.2)
    x=0;             % initial x
    y=2;             % initial y
    x_eval=1;        % at what point we have to evaluate
    n=(x_eval-x)/h; % Number of steps
    x4(1)=x;
    y4(1)=y;
    fprintf(' At x(%.2f) solution of y is %f\n',x,y)
    for i=1:n
    %RK4 Steps
       k1=h*double(f(x,y));
       k2=h*double(f((x+h/2),(y+k1/2)));
       k3=h*double(f((x+h/2),(y+k2/2)));
       k4=h*double(f((x+h),(y+k3)));
       dx=(1/6)*(k1+2*k2+2*k3+k4);
       x=x+h;
       y=y+dx;
       x4(i+1)=x;
       y4(i+1)=y;
       fprintf(' At x(%.2f) solution of y is %f\n',x,y)

    end
  
    fprintf('\n\tThe solution using Runge Kutta 4 for h=%.2f at x(%.1f) is %f\n',h,x4(end),y4(end))
  
%%Plotting solution using Euler method
figure(1)
hold on
plot(x2,y2,'Linewidth',2)
plot(x4,y4,'Linewidth',2)
xlabel('x')
ylabel('y(x)')
title('Solution plot y vs. x')
legend('Euler Method','RK4 Method','Location','northeast')
grid on

%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%