# Custom fitting using equation from differential function

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Sabella Huang on 1 Jun 2022
Answered: Sabella Huang on 3 Jun 2022
Hello guys,
I have some data that need to be fit by the equation from differential equation solver. I want to obtain parameter from syms A and B as an answer. Here is my code that I try to solve.
syms t y(t) A B
eqn_ass = diff(y,t) == A-B*y;
cond_ass = y(0) == 0;
R_ass = dsolve(eqn_ass,cond_ass); % I want to use equation from here as my fitting equation to obtain parameter A and B

Bjorn Gustavsson on 1 Jun 2022
Edited: Bjorn Gustavsson on 1 Jun 2022
Use matlabFunction to make a function out of the symbolic solution, then fit that one numerically to your data:
R_a_fcn = matlabFunction(R_ass);
err_fcn = @(par,t,y,fcn) sum((y - fcn(par(1),par(2),t)).^2); % error-function
res_fcn = @(par,t,y,fcn) (y - fcn(par(1),par(2),t));
% mock up test-example
t = 0:31;
y = 12*(1-exp(-t/11));
% Fit with fminsearch:
PAR = fminsearch(@(par) err_fcn(par,t,y,R_a_fcn),[1 1]);
% Fit with lsqnonlin:
PAR2 = lsqnonlin(@(par) res_fcn(par,t,y,R_a_fcn),[1 1]);
% Fit with lsqcurvefit:
PAR3 = lsqcurvefit(@(par,t) R_a_fcn(par(1),par(2),t),[1 1],t,y);
% Compare:
plot(t,y)
hold on
plot(t,R_a_fcn(PAR(1),PAR(2),t),'--')
plot(t,R_a_fcn(PAR2(1),PAR2(2),t),'-.')
plot(t,R_a_fcn(PAR3(1),PAR3(2),t),':')
HTH

Alan Stevens on 1 Jun 2022
You equation has the analytical solution y = (A/B)*(1-exp(-B*t)). Use fminsearch to fit the parameters, or the curve fitting toolbox if you have it.

Sabella Huang on 3 Jun 2022
its work... thx for your help