We present a couple of examples to solve an initial value problem for the first order differential equation in normal form using standard Matlab subroutine.

function solveDifferentialEquation
%This function solves y'+y = x from starttime to endtime with y(0) = intialCondition

%Parameter initializations starttime = 0; endtime = 20; initialCondition = -2;

function yprime = DifferentialEquationFunction(x,y) yprime = y - x; end

[xVals,yVals] = ode45(@DifferentialEquationFunction,[starttime,endtime],initialCondition); plot(xVals,yVals); end %Function Complete



			function solveDifferentialEquation2
              % Function to solve the differential equation
              % of a particle in motion due to impact of an external sinusoidal
              % force ( yprime = F*sin(w*t) - C*y )
              % from a starttime to stoptime, where the intial y value is assigned as
              % initialCondition
%Parameter initializations n = 5; onesv = [1, 1.45, 1.34, 1.81, 3.5]; %acts as a scalar vector that scales the parameters by its %entries. C = 10.*onesv; F = 1.*onesv; w = 1.*onesv; %With scaled frequencies,w, the plots should show a trend of increasing spread with t5he 3rd plot as an exception since 1.34 defies the pattern.

starttime = 0; endtime = 20; initialCondition = 0; counter = 1;

%Loop to solve the ODE and plot five different graphs.
for indeX = 1:n [t,y] = ode45(@DifferentialEquationFunction2,[starttime,endtime],initialCondition); counter = counter + 1; figure(indeX) plot(t,y);
title('Solution to differential equation') xlabel('time') ylabel('y-value') end

function yprime = DifferentialEquationFunction2(t,y)      % Function defining the ODE yprime = F(counter)*sin(w(counter)*t) - C(counter)*y; end
end          %Function Complete

There is a Matlab subroutine, credited to John Polking, called odesolve.m