To solve this Boundary Value Problem (BVP) recall that the general solution for this type of derivative is,

Therefore, the equation becomes

From here, apply the boundary conditions to solve for the constants  and 

Thus resulting in the solution,

To solve this Boundary Value Problem (BVP) recall that the general solution for this type of derivative is,

Therefore, the equation becomes

From here, apply the boundary conditions to solve for the constants  and 

Thus resulting in the solution,

This statement is true by the Uniqueness Theorem.

 is twice differential equation in terms of  and .

Now, consider the energy integral

After performing integration by parts results in the following,

From here, the initial conditions and boundary conditions are applied.

Therefore,

which proves the wave equations has only one solution and thus is unique.

The conservation law written as a partial differential equation is found by applying the divergence theorem to the conservation equation.

The conservation equation is,

Now, recall the divergence theorem which states,

Thus, by substituting 

 for  results in,

From here, rewriting this equation to bring the derivative inside the integral along with substituting

,

and performing some algebraic operations results in,

After integrating over the domain  the partial differential equation that is found is,

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

The partial derivatives are:

Notice that each partial derivative contains two variables, thus this equation is a second order partial differential equation.

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

The partial derivatives are:

Notice that one of them partial derivative contains three variables, thus this equation is a third order partial differential equation.

When dealing with partial differential equations, there are phenomenons in the physical world that have specific equations related to them in the mathematical world.

Looking at the possible answer selections below, identify the physical phenomena each represents.

  is known as the heat equation.

 is known as the wave equation.

 is known as the Laplace equation.

 is known as the Poisson equation.

 is known as the biharmonic wave equation.

Therefore, the correct answer for the biharmonic wave equation is 

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

The partial derivatives are:

Notice that one of them partial derivative contains three variables, thus this equation is a third order partial differential equation.