Last week was a basic introduction to Matlab. This week you will learn how to use three built-in functions (laplace, ilaplace and dsolve).
Using the syms function

1. An expression can be represented using symbols with the syms function.
Let's study the following example:
syms x y % Declare two symbolic variables x and y
y = x^2 %Declare a function
z=diff(y) %Will calculate the derivative of y and store the value in z.

2. How to find the Laplace transform for an expression?
Example: Find the Laplace transform of sin(2t). From the table of Laplace transforms, the Laplace transform of sin(2t) is . Let us do the same calculation in Matlab.
syms t % Declare a symbolic variable t
F=laplace(sin(2*t)) %Calculate Laplace transform, store in F
pretty(F) % print the output in a more readable form

3. How to find the inverse Laplace transform for an expression?
Example: Find the inverse Laplace transform of From the previous example, the inverse Laplace transform is sin(2t). The following code will verify this calculation in Matlab.
syms s % Declare a symbolic variable s (the Laplace variable)
f=ilaplace(2/(s^2+4)) %Calculate inverse Laplace transform, store in F
pretty(f) % print the output in a more readable form

4. How to solve a differential equation in Matlab?
Example: Solve the differential equation , with initial conditions x(0)=1 and x'(0)= 0.
Matlab command:
dsolve('D2x+2*x=0', 'x(0) = 1, Dx(0) = 0')

Exercises:
1. Find the Laplace transforms of the following functions by hand and verify results using Matlab.
2. Find the inverse Laplace transform of the following functions by hand and verify results using Matlab.

1.
2. F(s)=
3. Solve the following differential equations by hand and verify results using Matlab.

with x(0) = 1, x' (0) = 0

with x(0) = 1, x' (0) = 0