MATLAB: How to find a 2 π periodic solution

Question

How to find a 2 π periodic solution for the Equation?; x" + 2 x' + x =cos(t)

Also I need to find the solution to: x" + 2 x' + x =cos(t), x(0)=x'(0)=0

finally I need to plot the graph. Thanks for you help

Answer 1

• The first approach is to use the Symbolic Math Toolbox (See documentation). In order to solve the differential equation x" + 2 x' + x =cos(t), you can use the function dsolve (See documentation for dsolve). For example,

>> syms x(t);  
>> sol = dsolve('D2x+2*Dx+x = cos(t)', 't');

For the one with boundary conditions x" + 2 x' + x =cos(t), x(0)=x'(0)=0, you can try

>> syms x(t);
>> sol2 = dsolve('D2x+2*Dx+x = cos(t)', 'x(0)=0', 'Dx(0)=0');

To plot the answer stored in sol2, for example on the interval [0,10], you can do the following

>> t = linspace(0,10,1000);
>> plot(t,eval(sol2))

Notice that sol2 is a symbolic variable (See documentation for symbolic objects) , and therefore it needs to be evaluated before being executed. The way to do it is to use eval (See documentation for eval).

• Alternatively, you may want to consider numerical ordinary differential equation solvers, such as ode45 (See Documentation). Take a look at Examples 2 and 3 in the documentation. Example 2 shows how to handle higher order terms and Example 3 illustrates how to deal with time-varying terms.