Suppose we want to find out whether a function $\phi(z,t)$ is a solution to the one-dimensional classical wave equation
$\frac{\partial^2\phi}{{\partial}z^2}-\frac{1}{v^2}\frac{\partial^2\phi}{{\partial}t^2}=0$
The variables are
In addition there is a parameter
This page offers a two-step process using Sage to answer the question.
If Sage finds one or more solutions for real $v$ for the above differential equation, then we know $\phi$ is a wave function. The value for $v$ must not depend on $z$ or $t$. If Sage does not find a solution, we don't know for sure either way.
Use $f$ for $\phi$ in the following discussion.
In Sage, $\frac{\partial^2f}{{\partial}z^2}$ is given by f.diff(z,z).
Start by defining one function of $f$, solve_for_wave_velocity() that attempts to solve the general wave equation for $v$.
Define a second function, try_solution(f,v), that, given symbolic expressions for $f$ and $v$, allows us to see whether substituting these expressions in the wave equation gives a tautology, i.e. that the left-hand-side is identically zero for the given definitions of $f$ and $v$. The value returned is a symbolic expression. Sometimes the result of substitution will need additional simplification to show that it is identically zero.
Is $f(z,t)=sin(z+t)$ a solution to the wave equation?
Is $f(z,t)=sin(zt)$ a solution to the wave equation?
The expression for $v$ returned by solve_for_wave_velocity() for Example 2 depends on $z$ and $t$. That suggests this $f$ is not a wave function. In this case, it is not helpful to check the result with try_solution(f,v).
Is $f(z,t)=e^{\pi{z}-17t}$ a solution to the wave equation?
Since the solution for v is a constant and the result checks, Example 3 with $v=\frac{17}{\pi}$ is a solution to the wave equation.
Start with a known wave function. Change one of the constants in an attempt to "break" the function so that it is no longer a solution. Run the tests and find out the change created a new wave function.
So the second definition in Example 4 is also a wave function. How did that happen? Here are a few calculations.
See Week 1 content from the online course, QMSE01 Quantum Mechanics for Scientists and Engineers.
There is a matching textbook from Cambridge University Press, Quantum Mechanics for Scientists and Engineers.
The general solution to the one-dimensional wave equation, in the form $\phi(z,t)=F(z+ct)+G(z-ct)$, is provided by d´Alembert's formula.