what is i in the answer?

Hello,

I don't know what is i in the answer of my script. Complex number? how can I simplify it? I need to get rid of i.

syms x(t) s
Dx = diff(x(t),t);
laplace(sin(t)*Dx,t,s)

ans =

- (s*laplace(x(t), t, s - 1i)*1i)/2 + (s*laplace(x(t), t, s + 1i)*1i)/2 - laplace(x(t), t, s - 1i)/2 - laplace(x(t), t, s + 1i)/2

Thank you for helping me

[Stephen23]

"what is i in the answer?"

https://en.wikipedia.org/wiki/Imaginary_unit

[Si So]

but why should I have imaginary number here?

[Walter Roberson]

What good would it do you to get rid of the imaginary number?

You can perhaps get to something like

int(((t-s)*laplace(x(T),T,t-s)+x(0))/(t^2+1),t,0,s)

but then what? You are not going to be able to get rid of the unresolved laplace()

[Si So]

But, why MATLAB does not solve the final laplace?

[Walter Roberson]

How could it possibly solve the final laplace? You are dealing with the unknown function x(t) !

[Si So]

but in mathematics, it can be defined:

laplace{x(t)}=X(s)

I am solving an ODE by laplace method. So x(t) is the variable of differential equation.

[VBBV]

@Si So you could simplify the equation and then use lapplace term by term

syms x(t) s
Dx = diff(x(t),t);
L = laplace(laplace(sin(t),t,s)*Dx,t,s)

L = 

[Paul]

@VBBV

What is the mathematical justifcation for this solution?

syms x(t) s
Dx = diff(x(t),t);

Consider the inner term

(laplace(sin(t),t,s)*Dx)

ans = 

What does it mean to mutiply of function of t with a function of s (in the context of this problem)?

Is there a reference for a property of the Laplace transform that says ( * means multiplication)?

L(f(t)*g(t)) = L(f(t))*g(t)

%L = laplace(laplace(sin(t),t,s)*Dx,t,s)