# Partial Differential Equations : Derivatives from Conservation Laws

## Example Questions

### Example Question #1 : Derivatives From Conservation Laws

What is the conservation law written as a partial differential equation?

Explanation:

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,

### Example Question #2 : Derivatives From Conservation Laws

What is the order of the following partial differential equation.

Linear

Second Order

Third Order

First Order

Quasi Linear

Second Order

Explanation:

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.

### Example Question #3 : Derivatives From Conservation Laws

What is the order of the following partial differential equation.

Quasi Linear

Linear

Second Order

Third Order

First Order

Third Order

Explanation:

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.

### Example Question #4 : Derivatives From Conservation Laws

Which of the following describes the physical phenomena that is the biharmonic wave equation?

Explanation:

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

### Example Question #5 : Derivatives From Conservation Laws

What is the order of the following partial differential equation.

Homogeneous

Non-homogenous

Third Order

First Order

Second Order

Third Order

Explanation:

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.