# How do I solve an initial value ODE problem in MATLAB? Updated for MATLAB 2020a

Many students ask me how do I do this or that in MATLAB.  So I thought why not have a small series of my next few blogs do that.  In this blog, I show you how to solve an initial value ordinary differential equation.

• The MATLAB program link is here.
• The HTML version of the MATLAB program is here.
• DO NOT COPY AND PASTE THE PROGRAM BELOW BECAUSE THE SINGLE QUOTES DO NOT TRANSLATE TO THE CORRECT SINGLE QUOTES IN MATLAB EDITOR.  DOWNLOAD THE MATLAB PROGRAM INSTEAD

%% HOW DO I DO THAT IN MATLAB SERIES?
% In this series, I am answering questions that students have asked
% me about MATLAB. Most of the questions relate to a mathematical
% procedure.

%% TOPIC
% How do I solve an initial value ordinary differential equation?

%% SUMMARY

% Language : Matlab 2020b;
% Authors : Autar Kaw;
% Mfile available at
% http://nm.mathforcollege.com/blog/ode_initial_updated.m;
% Last Revised : December 22 2020;
% Abstract: This program shows you how to solve an
% initial value ordinary differential equation.
clc
clear all

%% INTRODUCTION

disp(‘ABSTRACT’)
disp(‘ This program shows you how to solve’)
disp(‘ an initial value ordinary differential equation’)
disp(‘ ‘)
disp(‘AUTHOR’)
disp(‘ Autar K Kaw of https://autarkaw.wordpress.com‘)
disp(‘ ‘)
disp(‘MFILE SOURCE’)
disp(‘ http://nm.mathforcollege.com/blog/ode_initial_updated.m‘)
disp(‘ ‘)
disp(‘LAST REVISED’)
disp(‘ Dec 22 2020’)
disp(‘ ‘)

%% INPUTS
% Solve the ordinary differential equation 3y”+5y’+7y=11exp(-x)
% Define x as a symbol
% Define y(x)n as a symbol also
syms x y(x)
%The ODE
Dy=diff(y);
ode_eqn=3*diff(y,x,2)+5*diff(y,x,1)+7*y == 11*exp(-13*x);
% The initial conditions
iv_1=Dy(0)==17;
iv_2=y(0)==19;
% The value at which y is sought at
xval=2.0;
%% DISPLAYING INPUTS

disp(‘INPUTS’)
func=[‘ The ODE to be solved is ‘ char(ode_eqn)];
disp(func)
iv_explain=[‘ The initial conditions are %s’ char(iv_1) ‘ ‘ char(iv_2)];
disp(iv_explain)
fprintf(‘ The value of y is sought at x=%g’,xval)
disp(‘ ‘)

%% THE CODE
conds=[iv_1 iv_2];
% Finding the solution of the ordinary differential equation
soln=dsolve(ode_eqn,conds);
soln=simplify(soln);
% vpa below uses variable-precision arithmetic (VPA) to compute each
% element of soln to 5 decimal digits of accuracy
soln=vpa(soln,5);

%% DISPLAYING OUTPUTS
disp(‘ ‘)
disp(‘OUTPUTS’)
output=[‘ The solution to the ODE is ‘ char(soln)];
disp(output)
value=subs(soln,x,xval);
fprintf(‘ The value of y at x=%g is %g’,xval,value)
disp(‘ ‘)

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• the textbooks on
• the Massive Open Online Course (MOOCs) available at

## Author: Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.