Differential Equations on Matlab

Let's talk about differential equations on Matlab.

First things first, what are differential equations? Well, they're a type of mathematical equation where an unknown function y(x) depends on its derivatives with respect to one or more independent variables. So let's say we have this differential equation: $$ y''(x)+y'(x)=0 $$ In this differential equation, the function y(x) is the unknown while its derivatives, y'(x) and y''(x), are known. Solving the differential equation means finding the function y(x).

Now, let me give you a practical example of how to do this on Matlab.

Consider this differential equation.

$$ y''(x)+y'(x)=0 $$

First, we need to define the symbol of the unknown function y(x) using the syms function.

syms y(x)

Next, we define the differential equation y''(x)+y'(x)=0 in the variable eqz, using the differentiation function diff()

For instance, we can write the first derivative y'(x) as diff(y,x,1) and the second derivative y''(x) as diff(y,x,2).

eqz = diff(y,x,2) + diff(y,x,1) == 0

Note. When defining the expression of the differential equation, use the comparison operator "==" to indicate the equal symbol "=" of the differential equation.

Finally, we can solve the differential equation using the function dsolve()

dsolve(eqz)

The function dsolve() calculates the general solution of the differential equation, which will be of the form C1+C2*exp(-x), where C1 and C2 are two arbitrary constants, and exp() is the exponential function.

C1+C2*exp(-x)

Therefore, the solution of the differential equation is

$$ y(x) = c_1 + c_2e^{-x} $$

So there you have it. By following this simple procedure, you can calculate the general solution of a homogeneous or non-homogeneous differential equation of any order using Matlab.