December 10, 2019, 11:10:26 am

### AuthorTopic: Substantial Differential Equation.  (Read 444 times)

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#### LucaKing

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##### Substantial Differential Equation.
« on: November 27, 2019, 12:29:16 pm »
0
Hi everyone!

My previous post posted on the QCE Methods forum was posted regarding my Math Assignment, which I have progressed in, but I am stuck on the one final part of my assignment.

I have a mammoth differential equation to solve that I have no idea how to solve and how to start this process.

Here it is

$\frac{100^2}{x^2}\cdot \left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4\cdot \left(1-\frac{1}{0.0019\left(\sqrt{2x^2}+6.6\right)^3}\right)^4\cdot \:132$

Thanks again!
Luca.

#### Tau

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##### Re: Substantial Differential Equation.
« Reply #1 on: November 27, 2019, 12:39:55 pm »
+1
Hi everyone!

My previous post posted on the QCE Methods forum was posted regarding my Math Assignment, which I have progressed in, but I am stuck on the one final part of my assignment.

I have a mammoth differential equation to solve that I have no idea how to solve and how to start this process.

Here it is

$\frac{100^2}{x^2}\cdot \left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4\cdot \left(1-\frac{1}{0.0019\left(\sqrt{2x^2}+6.6\right)^3}\right)^4\cdot \:132$

Thanks again!
Luca.

It looks like your missing an equality in this.... Otherwise it’s just an expression and there’s no DE to solve...
2018: Methods [41~], Physics [46~]
2019: Specialist, Algorithmics (HESS), UMEP Mathematics Extension [First Class Honours (H1)], English

#### LucaKing

• Posts: 8
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##### Re: Substantial Differential Equation.
« Reply #2 on: November 27, 2019, 12:54:09 pm »
0
It looks like your missing an equality in this.... Otherwise it’s just an expression and there’s no DE to solve...

Oh! Sorry, here:

$Productivity=\frac{100^2}{x^2}\cdot \left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4\cdot \left(1-\frac{1}{0.0019\left(\sqrt{2x^2}+6.6\right)^3}\right)^4\cdot \:\:132$

For my assignment, this is equation is needed to be differentiated in order to find the total productivity of a single tree surrounded by eight others in a 3x3 area. Not 100% sure if the productivity should be assigned P or 0. Here is my assignment sheet: https://imgur.com/gallery/AK88ZVv

This is just for context: I have figured out A+B, my orchard design, the equations for productivity and I am at the final stage of deriving the equation stated above for the total productivity of one tree surrounded by eight. Once that is found, I can find out the productivity of the sides and the corners as they are going to be slightly different than the center part to find the total productivity of the orchard!

Thanks again
Luca
« Last Edit: November 27, 2019, 02:45:18 pm by LucaKing »

#### RuiAce

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##### Re: Substantial Differential Equation.
« Reply #3 on: November 27, 2019, 03:45:54 pm »
+1
"Differential equation" is a misnomer here I'd say, since what you're trying to do is just maximise productivity here; hence you're solving an optimisation problem.

I'm not too sure why the $\frac{100^2}{x^2}$ term is meant to represent here, but I'll trust that you understand your design well enough. The $\sqrt{2x^2}$ part makes sense to me though.

So, to help combat the actual differentiation, here are my suggestions:

1. Firstly, because you're just assuming that $x$ is a distance here, you should be safe to assume that $x\geq 0$. Hence the $\sqrt{2x^2}$ term can be simplified to $\sqrt{2}x$. (Here, the $\sqrt{2}$ has now basically become the coefficient of that $x$ term.)

2. If you don't mind using the quotient rule a little, you may consider trying to simplify $1 - \frac{1}{0.0019(x+6.6)^3}$ into one common denominator. Same goes for $1 - \frac{1}{0.0019(\sqrt{2}x+6.6)^3}$.

3. Since the two powers happen to coincide, you may wish to consider merging them under the common power into this:
$\left( \left(1 - \frac{1}{0.0019(x+6.6)^3}\right)\left(1 - \frac{1}{0.0019(\sqrt2 x+6.6)^3}\right) \right)^4 .$
From here, you could just expand the brackets inside the root. Alternatively, you could combine this wth point 2, and expand after combining the fractions.

4. The $\frac{1}{x^2}$ term can technically also be moved in under the power, if you rewrite it as $\left(\frac{1}{\sqrt{x}}\right)^4$. Personally not a fan of this suggestion though.

5. You may instead wish to just use the generalised product rule to help you out. The version for three terms is mentioned on the Wikipedia page for the product rule.

6. This stretches the boundaries of maths methods a lot, because technically speaking it requires something from spesh. But if you're willing to go the extra mile in research (and if your assignment lets you do so...), you may wish to consider researching a technique called logarithmic differentiation. It's extra time trying to learn the mechanics of it, but it can help reduce a lot of the product rule and chain rule computations required.

In saying this: I have a bad feeling that you may be unable to solve for $x$, when you set $\frac{dp}{dx} = 0$ (where $p = productivity$). I could be assuming things too quickly, but for all I know, it may be a problem that falls outside the limitations of basic algebra.

Of course, you may also wish to consider a simpler design. But that's not something I would recommend for, nor recommend against.

#### LucaKing

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##### Re: Substantial Differential Equation.
« Reply #4 on: November 27, 2019, 06:12:18 pm »
0
"Differential equation" is a misnomer here I'd say, since what you're trying to do is just maximise productivity here; hence you're solving an optimisation problem.

I'm not too sure why the $\frac{100^2}{x^2}$ term is meant to represent here, but I'll trust that you understand your design well enough. The $\sqrt{2x^2}$ part makes sense to me though.

So, to help combat the actual differentiation, here are my suggestions:

1. Firstly, because you're just assuming that $x$ is a distance here, you should be safe to assume that $x\geq 0$. Hence the $\sqrt{2x^2}$ term can be simplified to $\sqrt{2}x$. (Here, the $\sqrt{2}$ has now basically become the coefficient of that $x$ term.)

2. If you don't mind using the quotient rule a little, you may consider trying to simplify $1 - \frac{1}{0.0019(x+6.6)^3}$ into one common denominator. Same goes for $1 - \frac{1}{0.0019(\sqrt{2}x+6.6)^3}$.

3. Since the two powers happen to coincide, you may wish to consider merging them under the common power into this:
$\left( \left(1 - \frac{1}{0.0019(x+6.6)^3}\right)\left(1 - \frac{1}{0.0019(\sqrt2 x+6.6)^3}\right) \right)^4 .$
From here, you could just expand the brackets inside the root. Alternatively, you could combine this wth point 2, and expand after combining the fractions.

4. The $\frac{1}{x^2}$ term can technically also be moved in under the power, if you rewrite it as $\left(\frac{1}{\sqrt{x}}\right)^4$. Personally not a fan of this suggestion though.

5. You may instead wish to just use the generalised product rule to help you out. The version for three terms is mentioned on the Wikipedia page for the product rule.

6. This stretches the boundaries of maths methods a lot, because technically speaking it requires something from spesh. But if you're willing to go the extra mile in research (and if your assignment lets you do so...), you may wish to consider researching a technique called logarithmic differentiation. It's extra time trying to learn the mechanics of it, but it can help reduce a lot of the product rule and chain rule computations required.

In saying this: I have a bad feeling that you may be unable to solve for $x$, when you set $\frac{dp}{dx} = 0$ (where $p = productivity$). I could be assuming things too quickly, but for all I know, it may be a problem that falls outside the limitations of basic algebra.

Of course, you may also wish to consider a simpler design. But that's not something I would recommend for, nor recommend against.

Thanks again Rui, your help is greatly appreciated!

If you may be wondering, the $\frac{100^2}{x^2}$ represents the space taken up by a tree where $100^2$ is the area and $x^2$ is the area per tree. I'll get to solving it right away, and I sincerely hope you are incorrect about this problem potentially having no solution.

Again, can't thank you more enough.
Have a wonderful rest of your day
Luca.

#### LucaKing

• Posts: 8
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##### Re: Substantial Differential Equation.
« Reply #5 on: November 29, 2019, 02:18:33 pm »
0
"Differential equation" is a misnomer here I'd say, since what you're trying to do is just maximise productivity here; hence you're solving an optimisation problem.

I'm not too sure why the $\frac{100^2}{x^2}$ term is meant to represent here, but I'll trust that you understand your design well enough. The $\sqrt{2x^2}$ part makes sense to me though.

So, to help combat the actual differentiation, here are my suggestions:

1. Firstly, because you're just assuming that $x$ is a distance here, you should be safe to assume that $x\geq 0$. Hence the $\sqrt{2x^2}$ term can be simplified to $\sqrt{2}x$. (Here, the $\sqrt{2}$ has now basically become the coefficient of that $x$ term.)

2. If you don't mind using the quotient rule a little, you may consider trying to simplify $1 - \frac{1}{0.0019(x+6.6)^3}$ into one common denominator. Same goes for $1 - \frac{1}{0.0019(\sqrt{2}x+6.6)^3}$.

3. Since the two powers happen to coincide, you may wish to consider merging them under the common power into this:
$\left( \left(1 - \frac{1}{0.0019(x+6.6)^3}\right)\left(1 - \frac{1}{0.0019(\sqrt2 x+6.6)^3}\right) \right)^4 .$
From here, you could just expand the brackets inside the root. Alternatively, you could combine this wth point 2, and expand after combining the fractions.

4. The $\frac{1}{x^2}$ term can technically also be moved in under the power, if you rewrite it as $\left(\frac{1}{\sqrt{x}}\right)^4$. Personally not a fan of this suggestion though.

5. You may instead wish to just use the generalised product rule to help you out. The version for three terms is mentioned on the Wikipedia page for the product rule.

6. This stretches the boundaries of maths methods a lot, because technically speaking it requires something from spesh. But if you're willing to go the extra mile in research (and if your assignment lets you do so...), you may wish to consider researching a technique called logarithmic differentiation. It's extra time trying to learn the mechanics of it, but it can help reduce a lot of the product rule and chain rule computations required.

In saying this: I have a bad feeling that you may be unable to solve for $x$, when you set $\frac{dp}{dx} = 0$ (where $p = productivity$). I could be assuming things too quickly, but for all I know, it may be a problem that falls outside the limitations of basic algebra.

Of course, you may also wish to consider a simpler design. But that's not something I would recommend for, nor recommend against.

Hi again!
I have differentiated the equation, however I would like to check if it is correct.
$\left(\frac{-2640000}{x^3}\right)\cdot \left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4\cdot \left(1-\frac{1}{0.0019\left(\sqrt{2}x+6.6\right)^3}\right)^4+\left(\left(\frac{1320000}{x^2}\right)\cdot \left(\frac{3}{0.0019\left(x+6.6\right)^4}\left(1-1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)\right)^3\cdot \left(1-\frac{1}{0.0019\left(\sqrt{2}x+6.6\right)^3}\right)^4\right)+\left(\left(\frac{-2640000}{x^3}\right)\cdot \left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4\cdot \left(\frac{3\sqrt{2}}{0.0019\left(\sqrt{2}x+6.6\right)^4}\left(1-\frac{1}{0.0019\left(\sqrt{2}x+6.6\right)^3}\right)\right)\:=0\:\:\:\right)$

First off, there are some formatting issues within the formula with the double brackets and the bracket around the 0, however, I believe the numbers inside are correct.

I've written the equation in the format of: $\frac{dy}{dx}=f'\left(x\right)g\left(x\right)h\left(x\right)+f\left(x\right)g′\left(x\right)h\left(x\right)+f\left(x\right)g\left(x\right)h′\left(x\right)$

With $f=\frac{1320000}{x^2}$ , $g=\left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4$ and $h=\left(1-\frac{1}{0.0019\left(x+6.6\right)^3}\right)^4$

Would greatly appreciate it if someone confirms my differentiation or corrects it! I have tried to plot it within Desmos, and it has not worked, I tried to find a value in a graphing calculator and it did not like it.

You guys are my last hope!
Cheers, Luca.

#### Specialist_maths

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##### Re: Substantial Differential Equation.
« Reply #6 on: November 30, 2019, 07:42:56 am »
+1
If you may be wondering, the $\frac{100^2}{x^2}$ represents the space taken up by a tree where $100^2$ is the area and $x^2$ is the area per tree. I'll get to solving it right away, and I sincerely hope you are incorrect about this problem potentially having no solution.
You're assuming that the 1 hectare orchid is a square. Is there a reason for that?

Also, if this is your IA1 for Methods (QCE), you should also be using technology to solve the problem (syllabus, p30). I'd recommend using Desmos but you could use your graphics calculator if you have the software to capture screenshots. Graphing the productivity equation you listed shows a turning point around x=8.
Teacher of 2020 Seniors: Specialist Mathematics + Mathematical Methods
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#### LucaKing

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##### Re: Substantial Differential Equation.
« Reply #7 on: November 30, 2019, 11:11:33 am »
0
You're assuming that the 1 hectare orchid is a square. Is there a reason for that?

Also, if this is your IA1 for Methods (QCE), you should also be using technology to solve the problem (syllabus, p30). I'd recommend using Desmos but you could use your graphics calculator if you have the software to capture screenshots. Graphing the productivity equation you listed shows a turning point around x=8.

Hey!

Although it is not written on the task sheet, we can only fit our orchard into a 100mx100m as I could have created a 1mx10000m field and had only two trees effecting one (not including the corners) to achieve a super high orchard productivity. Whilst my lecturer was going over it in class, it was noticed and a 100mx100m area was enforced.

Yes, this is my IA1, which is one of two assignments, including tests, this year that I have received and I have been using Desmos and graphing calculators to solve the equation. Last night, I realized I had been interpreting the graph wrong to find the correct answer and have now found the correct value

Appreciate the help of everyone here.
Thankfully, the contents of the assignment are finished and I just now have to write up the final piece.
Hopefully this is the end of this thread.
Luca.