Euler's method in MATLAB: code doesn't work - matlab

Question

For my computing course, I am given the following function:
y'(x) = -8y(x) + 0.5x + 1/16 and an initial value of y(0)=2.
Now, I am asked to solve this equation by using Euler's method, in MATLAB.
My code should give an output of 2 arrays: xar and yar, in which I see the x-values vs. the y-values, however, if I run my code, it says: "undefined variable x". Here's my code:
function [xar,yar] = Euler(a,b,ybouco,N)
% a is the lower limit
% b is the upper limit
% ybouco is the initial value
% N is the number of intervals
a=0;
b=3;
ybouco=2;
N=10;
h=(b-a)/N;
T=a:h:b;
y(1)=ybouco;
f = #(x) -8*y(x) + 0.5x + (1/16);
y(x) = 2*exp(-8*x)+(1/16)*x;
for i = 1:N
y (i+1) = y(i)+h*f(T(i));
end
end
Can someone explain what is wrong with my code??

Answer 1

First of all, note that assigning argument parameters in the function block is wrong! (i.e. a,b,ybouco and N) should be passed through argument by calling the function. There is no use in writing arguments to be assigned by user beside assigning them in the script manually.
One way is to call the function and assign the value in the command window like below:
[x,y]=Euler(0,3,2,10)
where a=0, b=3, ybouco=2 and N=10 was passed to the function as an input and x and y returned by the function as output.
Plus, when you are solving an ODE numerically it means you do not know y analytically.
So you should omit the assigning part of the code and do a little change like below:
function [xar,yar] = Euler(a,b,ybouco,N)
h=(b-a)/N;
T=a:h:b;
y(1)=ybouco;
for i = 1:N
f(i) = -8*y(i) + 0.5*T(i) + (1/16);
y(i+1) = y(i)+h*f(i);
end
xar=T;
yar=y;
end
Then by calling the function in the command window, you will get the following results:
x =
Columns 1 through 8
0 0.3000 0.6000 0.9000 1.2000 1.5000 1.8000 2.1000
Columns 9 through 11
2.4000 2.7000 3.0000
y =
Columns 1 through 8
2.0000 -2.7813 3.9575 -5.4317 7.7582 -10.6627 15.1716 -20.9515
Columns 9 through 11
29.6658 -41.1533 58.0384
You can also plot the result and get the following graph:
If you increase N from 10 to 100 you will have more accurate results and a smooth graph like below:

Answer 2

The error message is because you have an assignment
y(x) = 2*exp(-8*x)+(1/16)*x;
where x is not defined. The x in y(x) indexes into the array y.
Maybe you intended to write
y = #(x) 2*exp(-8*x)+(1/16)*x;
to define an anonymous function. But that would clash with the array y you have already defined. Maybe just delete this line?
Also,
h=(b-a)/N;
T=a:h:b;
can be better written as
T = linspace(a,b,N);