Solving for a function after replacing and substituting in an expression

Question

I am trying to isolate $X(s)$ in the following expression:

e1 = LaplaceTransform[m x''[t] + c x'[t] + k x[t] == HeavisideTheta[t], t, s]
e1 /. LaplaceTransform[x[t], t, s] -> X[s]
Solve[e1, X[s]]

But Solve returns an empty array. What is wrong here?

Answer 1

Doing e1 /. LaplaceTransform[x[t], t, s] -> X[s] doesn't replace anything in the expression e1. It makes a substitution/replacement according to some rule and displays the result, but e1 remains unchanged. This might be a bit confusing due to the replacing name of the function. But consider its operator form:

`ReplaceAll[rule][expression]`

ReplaceAll[rule] is an operand acting on expression. It has the same structure like every other MMA command, e.g. Det: Det[mat] computes the determinant of mat and displays the result; it does not change mat into its determinant, mat is unchanged. The same reasoning applies to ReplaceAll.

So, if one does

e2 = e1 /. LaplaceTransform[x[t], t, s] -> X[s]
Solve[e2, X[s]]

the result will be

NOTE: In the above I made some general statements for the sake of illustration. However, it's important to emphasize that there are commands that actually change the expression they act on, e.g. AppendTo. See this post for a discussion.