System of 2 differential equations

I'm trying to express the following two differential equations in matlab in a form that will allow me to input them into a solver such as ode45. How would I input the following equations into matlab:

dC/dt = (-exp((-10/(T+273)))*C

dT/dt = 1000*(exp((-10)/(T+273))*C-10*(T-20)

where C and T are concentration and temperature (dependent variables) and t is time (independent variable).

Answer 1

[Star Strider]

The code you posted has errors, specifically unbalanced parentheses. I edited your equations to correct this, however I am not absolutely certain that my changes produce the correct result.

Try this:

% % % MAPPING: y(1) = C,  y(2) = T
CTfcn = @(t,y) [-exp(-10/(y(2)+273))*y(1); 1000*(exp(-10/(y(2)+273))*y(1)-10*(y(2)-20))];

tv = linspace(0, 1E-3);
ic = [0, 0];
[T,Y] = ode45(CTfcn, tv, ic);

figure
plot(T,Y)
grid
legend('C', 'T')

The first row of ‘CTfcn’ is ‘C’, the second is ‘T’.

I also do not know what the initial conditions should be. I use [0,0] here, however they do not appear to have a significant effect on the ‘T’ result, regardless what their values are. Please provide the correct initial conditions, and please check to be certain your equations are correct as posted, and as I coded them.

[Filip Jackowski]

my apologies on the late reply, but what should the order be for the initial conditions? if @T=15, C=1, what would be the order for the "ic" term? would it be [15, 1] or [1, 15]?

[Star Strider]

No worries.

It should be:

ic = [1, 15];

since the first row is ‘C’ and the second row is ‘T’.

The complete code (with a slightly revised figure call using subplot and a changed ‘tv’ vector so that both integrated variables are clearly visible) is now:

% % % MAPPING: y(1) = C,  y(2) = T
CTfcn = @(t,y) [-exp(-10/(y(2)+273))*y(1); 1000*(exp(-10/(y(2)+273))*y(1)-10*(y(2)-20))];
tv = linspace(0, 5E-4);
ic = [1 15];
[T,Y] = ode45(CTfcn, tv, ic);
figure
subplot(2,1,1)
plot(T,Y(:,1))
title('C(t)')
xlim([min(tv) max(tv)])
grid
subplot(2,1,2)
plot(T,Y(:,2))
title('T(t)')
xlim([min(tv) max(tv)])
grid

[Filip Jackowski]

thank you so much! pardon the nooby question, but how would i put them on the same graph and have two sets of y axes?

[Star Strider]

My pleasure!

No worries. Use the yyaxis (link) function:

% % % MAPPING: y(1) = C,  y(2) = T
CTfcn = @(t,y) [-exp(-10/(y(2)+273))*y(1); 1000*(exp(-10/(y(2)+273))*y(1)-10*(y(2)-20))];
tv = linspace(0, 5E-4);
ic = [1 15];
[T,Y] = ode45(CTfcn, tv, ic);
figure
yyaxis left
plot(T,Y(:,1))
yyaxis right
plot(T,Y(:,2))
grid
legend('C', 'T')

The yyaxis function was introduced in R2016a. (There was a previous function called plotyy (link) that produced a second y-axis. See the documentation for it if you have R2015b or earlier.)

[Filip Jackowski]

You're a lifesaver man. Many thanks!

[Star Strider]

As always, my pleasure!