# Simplify the follwoing Equation

• andrey21
In summary: Yes, that looks correct. In summary, the equation for sound waves in a pipe of varying cross-section can be described by the equation V2 d/dx (1/A dAu/dx) = d2u/dt2, where A = 0.2+0.3x. To simplify the equation, A is substituted in and then the quotient rule is applied. The simplified equation is V2 d/dx ((0.3du/dx-(0.3)2u)/(0.2+0.3x)2 + du/dx).
andrey21
Sounds waves in a pipe of varying cross-section are described by the equation

V2 d/dx (1/A dAu/dx) = d2u/dt2

Where A = 0.2+0.3x

So first I substituted A into the equation:

V2 d/dx (1/(0.2+0.3x) d(0.2+0.3x)u/dx) = d2u/dt2

V2 d/dx (0.3u/(0.2+0.3x) du/dx) = d2u/dt2This is as far as I can get, any help would be fantastic.

I suppose this is V² d/dx (1/A d(Au)/dx) = d²u/dt², true? Then expand first before substituting, which gives
d²u/dx² + d(ln A)/dx du/dx + d²(ln A)/dx² u = 1/V² d²u/dt².
Now substitute to get
d²u/dx² + 1/(x + 2/3) du/dx - 1/(x + 2/3)² u = 1/V² d²u/dt², which I think is about as simple as it gets.

(PS: This has a solution u(x,t) = [ C1 x² (1 + x)/(2 + 3 x) + C2 (1 + (2 + 3 x)²)/(2 + 3 x) + C3 (
(1 - (2 + 3 x)²))/(2 + 3 t) ] [ 1/2 C1 t² V² + C4 + t C5 ], where C1 to C5 are arbitrary constants. But that may not be the solution you are looking for.)

Last edited:
So the equation you have is:
$$v^{2}\frac{\partial}{\partial x}\frac{1}{A(x)}\frac{\partial}{\partial x}(A(x)u)=\frac{\partial^{2}u}{\partial t^{2}}$$
Use:
$$\frac{\partial}{\partial x}(Au)=A\frac{\partial u}{\partial x}+0.3u$$
Likewise for the 1/A term too.

Thank you for the replies, hunt_mat could u please explain your post a little more please I am confused

I will explain the second point:
$$\frac{\partial}{\partial x}(A(x)u)=A\frac{\partial u}{\partial x}+u\frac{\partial A}{\partial x}=A\frac{\partial u}{\partial x}+0.3u$$

Ok using what you have said I have obtained:

V2 d/dx (0.3u/0.2+0.3x + du/dx)

V2 d/dx (0.15u+ u/x + du/dx)

Is this correct??

You have to use the quotient rule which is:
$$\frac{d}{dx}\left(\frac{X}{A}\right) =\frac{A\frac{dX}{dx}-X\frac{dA}{dx}}{A^{2}}$$
Where:
$$X=\frac{\partial }{\partial x}(A(x)u))$$

Ok so I should use the quotient rule on:

0.3u/0.2+0.3x

(0.2+0.3x) (0.3) - 0.3u (0.3) / (0.2+0.3x)2

0.6 +0.9x -0.9u / (0.2+0.3x)2

am I on the right track??

Not quite, you should have:
(0.3du/dx-(0.3)^2u)/(0.2+0.3x)2

V2 d/dx ((0.3du/dx-(0.3)2u)/(0.2+0.3x)2 + du/dx)

## 1. What does it mean to "simplify" an equation?

Simplifying an equation involves reducing it to its most basic form by combining like terms, eliminating unnecessary elements, and solving for the desired variable.

## 2. Why is it important to simplify equations?

Simplifying equations allows for easier understanding and analysis of mathematical relationships. It also helps to identify patterns and make predictions based on the equation.

## 3. How do I know when an equation is simplified?

An equation is considered simplified when it cannot be reduced any further and all variable terms are on one side and constant terms on the other.

## 4. Can all equations be simplified?

Not all equations can be simplified. Some equations, such as quadratic equations, require more complex methods to solve and cannot be simplified to a basic form.

## 5. What are some common techniques for simplifying equations?

Some common techniques for simplifying equations include combining like terms, distributing and factoring, and using the order of operations. It may also involve using inverse operations to isolate the desired variable.

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