How to determine an x value at a specific y value?

I need help finding the value of t and B when S=0.001. I got my plot to work but not sure how to write my code to find these values at a specific S value.

t = [0:1:50]; 
Y0 = [0.05 5]; % initial values
[tout, Yout] = ode15s(@func2,t,Y0)

plot(tout,Yout)
xlabel('t (Time)')
ylabel('B and S ')
legend('B','S')
title('ODE15s')

function dy = func2(t,x)
kr = 0.4;
k = 0.003;

dy = zeros(2,1);
dy(1) = (kr.*x(1).*x(2))./(k+x(2))
dy(2) = -(0.75.*kr.*x(1).*x(2))./(k+x(2))

end

Answer 1

[Star Strider]

Try this:

t = [0:1:50]; 
Y0 = [0.05 5]; % initial values
[tout, Yout] = ode15s(@func2,t,Y0);

[Smin,Sminidx] = min(Yout(:,2))
idxrng = 1:Sminidx;
Sval = 0.001;
Time_and_B_at_S = interp1(Yout(idxrng,2), [tout(idxrng) Yout(idxrng,1)], Sval)
txtstr = sprintf('\\leftarrowAt S = %.3f\n          t = %.3f\n          B = %.3f',Sval, Time_and_B_at_S);

figure
plot(tout,Yout)
xlabel('t (Time)')
ylabel('B and S ')
legend('B','S')
title('ODE15s')
text(Time_and_B_at_S(1), Sval, txtstr, 'HorizontalAlignment','left', 'VerticalAlignment','middle')

function dy = func2(t,x)
kr = 0.4;
k = 0.003;

dy = zeros(2,1);
dy(1) = (kr.*x(1).*x(2))./(k+x(2));
dy(2) = -(0.75.*kr.*x(1).*x(2))./(k+x(2));

end

Choose ‘Sval’ to be whatever value you want (within the limits of ‘S’).

EDIT — (12 Nov 2020 at 22:44)

I did not initially see that you specified a value for ‘S’ that you want to interpolate. (Was that added later?) I have changed my code to reflect thae requested ‘S’ value.

Answer 2

[Walter Roberson]

t = [0:1:50]; 
Y0 = [0.05 5]; % initial values
[tout, Yout] = ode15s(@func2,t,Y0)

tout = 51×1

0 1 2 3 4 5 6 7 8 9

●

Yout = 51×2

0.0500 5.0000 0.0748 4.9814 0.1116 4.9538 0.1666 4.9125 0.2488 4.8509 0.3713 4.7590 0.5541 4.6219 0.8268 4.4174 1.2334 4.1124 1.8400 3.6575

plot(tout,Yout)
xlabel('t (Time)')
ylabel('B and S ')
legend('B','S')
title('ODE15s')

targetS = 0.001;
[~, minidx] = min(abs(Yout(:,2)-targetS));

fprintf('Closest time to S = %g is %g with B = %g and S = %g\n', targetS, t(minidx), Yout(minidx,:));

Closest time to S = 0.001 is 17 with B = 6.71667 and S = 4.85264e-20

function dy = func2(t,x)
kr = 0.4;
k = 0.003;
dy = zeros(2,1);
dy(1) = (kr.*x(1).*x(2))./(k+x(2));
dy(2) = -(0.75.*kr.*x(1).*x(2))./(k+x(2));
end

This is not an especially good match. To get closer, I would recommend using an Event Function to detect

x(2) - targetS

for crossing 0 and non-terminal. And use the third output of ode45() to get the time of the event.