The other day a student came to ask me for help in solving a first order ordinary differential equation using the ode45 routine of MATLAB. To use ode45, one needs to be familiar with how the inputs are required by MATLAB.

To solve a first-order ODE by ode45 is straightforward.

The ODE given is given as

3*dy/dx+7*y^1.2=5*x^1.1, y(2)=13.

Find the value of y(19).

**Program without comments**

clc

clear all

dydx=@(x,y) (5*x^1.1-7*y^1.2)/3

[xx,yy]=ode45(dydx,[2,19],13);

n=length(yy);

yend=yy(n)

**Program with comments**

clc

clear all

% Assume that you are given a first-order differential equation

% 3*dy/dx+7*y^1.2=5*x^1.1, y(2)=13.

% Find the value of y(19).

% How would you solve it by using the ode45 MATLAB function?

% SOLUTION

% First you would need to rewrite ODE as dy/dx=f(x,y) form

% dy/dx=(5*x^1.1-7*y^1.2)/3, y(2)=13

% Define a variable of your choice and write what dy/dx is.

% @(x,y) means these are the independent and dependent variables in ODE

dydx=@(x,y) (5*x^1.1-7*y^1.2)/3

% Look at the ode45 help in MATLAB

% Left hand side yy vector is where you want the values stored.

% xx is the vector that will be chosen by MATLAB, not you, at which it will

% provide you the value of yy vector.

% Inputs to ode45 are the following.

% 1) dydx is the ODE in the form of Line 14.

% 2) [2,19] is the span of xx values. You can observe x=2 is where the

% initial condition is given and x=19 is the value at which you are

% seeking the y value at. These two inputs can be variables too.

% 3) Last input is value of the initial condition that is given at x=2.

% This input can be a variable too.

[xx,yy]=ode45(dydx,[2,19],13);

% Since last entry of yy vector would be where the xx span ends, and that

% is where you want to find the value of y

n=length(yy);

yend=yy(n)

________________________________________

This post is brought to you by

- Holistic Numerical Methods Open Course Ware:
- Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
- Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com

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