%% Differential Equations with MATLAB %% Using MATLAB to give a numerical solution to an ODE %% % The ODE is %% % % \$\$y' = 2y \,\mbox{--}\, 1,\ y(0)=1.\$\$ % % We use *ode45* to obtain the numeric solution. We have to define a % MATLAB function equal to the right side of the equation, which we can do % with an anonymous function. syms t f = @(t,y) 2.*y -1 %% % To solve and plot the approximate solution ya on the interval [0,1], % we give the command ode45(f, [0,1], 1) %% % The second argument is the interval and the last one is the value of the % solution at the left end of the interval. %% % This equation is linear, so it is easy to solve symbolically. dsolve('Dy = 2*y-1, y(0)=1','t') %% % We can make a table of values of the approximate and exact solutions. We % don't want an enormous table, so we only calculate at t=0,0.1,0.2,...,1. % The first column in the table is t, the second the value of the % approximate solution, and the third the value of the exact solution. [t,ya] = ode45(f,0:0.1:1,1); y = 1./2+1./2.*exp(2.*t); format long [t,ya,y] %% % The approximate and exact solutions agree to 6 decimal places. This is a % contrast with the result we had for the same equation when we used % Euler's method.