The question is asking which of the given functions is a solution to the 1D wave equation:

$2 \frac{\partial^2 u}{\partial x^2} = \frac{1}{18} \frac{\partial^2 u}{\partial t^2}.$

Let's go over the options step by step to check if they satisfy the wave equation.

Step 1: Simplifying the wave equation

The given equation can be rewritten as:

$\frac{\partial^2 u}{\partial x^2} = \frac{1}{36} \frac{\partial^2 u}{\partial t^2}.$

This is the 1D wave equation with a wave speed $c = \frac{1}{6}$. A solution to this equation typically has the form of a function that depends on $x - ct$ or $x + ct$ (i.e., a traveling wave solution).

Step 2: Analyzing each option

Let's check if each given option is a valid solution.

• Option (I): $u(x, t) = \sin(x + \sqrt{18}t) + \sin(x - \sqrt{18}t)$

This is a standard solution for the wave equation, with wave speed $c = \sqrt{18} = 6$. Thus, option (I) is a solution.

• Option (II): $u(x, t) = \sin(2x - 12t) + 10 \cos \left( \frac{1}{18} x + \frac{1}{3} t \right)$

This consists of two terms:

• The first term is a sinusoidal wave with wave speed $c = \frac{12}{2} = 6$, which matches the equation.
• The second term is a cosine wave with a different form and doesn't match the wave speed of the equation, making this a non-solution.

Thus, option (II) is not a solution.

• Option (III): $u(x, t) = x^2 - 12xt + 36t^2$

This is a polynomial in $x$ and $t$. When applying the second derivatives, this does not result in a valid wave equation solution, as the second derivatives will not satisfy the form of the wave equation.

Thus, option (III) is not a solution.

• Option (IV): $u(x, t) = (x - 36t)^{100}$

While this represents a traveling wave, the exponent 100 does not correspond to a typical solution for the wave equation, as higher powers like this tend to distort the wave speed or fail to satisfy the required form.

Thus, option (IV) is not a solution.

Conclusion:

The correct answer is Only (I).

Would you like more details on how I derived these?

Here are some related questions you might be interested in:

1. How do you find general solutions for wave equations?
2. What are the typical forms of solutions for partial differential equations?
3. How does the wave speed affect the form of the solution in wave equations?
4. Can polynomial expressions ever satisfy wave equations?
5. What types of functions satisfy second-order partial differential equations?

Tip: Always check for the form of the equation and the corresponding wave speed when solving wave equations.