Differential Equations: How they relate to Calculus.
This is my first mathematics paper. It is on how differential equations relate to calculus. Any constructive criticism by those in the field is encouraged. You may download it by clicking the link below. (It is a Microsoft Word document)
Differential Equations:How they relate to Calculus
by Joel Guttormson
December 16, 2008  Posted by Metro State Atheists  Metro State Atheists, science, Uncategorized  Advanced Calculus, Archibald, Calculus, Dennis Zill, differeniable, Differential Equations, Differentiation, Discoveries, discovery, Elements of Differential equations, Euler, First Fundemental Theorem of Calculus, Fraser, Gauss, GrattanGuinness, History, history of differential equations, history of math, Integratal, intergrable, James Stewart, Jason Socrates Bardi, John and James Bernoulli, John Bernoulli, L'Hospital, Leibniz, math, Mathematics, Newton, Patrick M. Fitzpatrick, Power Series, Proof, proof of the first fundemental theorem of calculus, Stroud and Booth, The Bernoulli Brothers, The Calculus Wars, Wilfred Kaplan
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Maybe useful for you all, in the following address
http://math152.wordpress.com/2008/11/09/dunhamstalkoneuler/
afasmt informs the discussion of Bernoulli integral related with Bernoulli differential equation that created by a new solver of nonlinear first order ODE by socalled Stable Modulation Technique.
Comment by rohedi  January 5, 2009
Thank you very much for this additional information.
Joel
President
Metro State Atheists
Comment by Metro State Atheists  January 5, 2009
@rohedi
After visiting to
http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34&start=20.
Okay, now I post here an analytic solution of
dy/dx + Py = Q,
Rohedi’s Formula for the above linear first order ODE is of form
y(x)=[y0Q/P]exp{P(xx0)} + Q/P
Now, suppose we have a problem 4dF/dt=32F, in which at t0=t(0)=0 , F0=F(0)=3, then we must
write this ODE as following
dF/dt + (1/2)F = 3/4 ,
Here, we have P=1/2 and Q=3/4, here y=F and x=t, x0=t0=0 and y0=F0=3. The exact solution created by using Rohedi’s Formula is of form
F(t) = [3 – 3/2]exp{t/2} + 3/2
or
F(t) = (3/2)[exp{t/2} + 1]
Please verify the above solution with the usual way.
Comment by Nadya Fermega  January 6, 2009
@Nadya Fermega
An interesting discussion. I am interested to verify the solution of 4dF/dt=32F created using Rohedi’s Formula.
Okay, let’s start by writting the linear ODE as following
dF/dt=3/4 – F/2
Separating variables of F and t gives
dF/(3/4 – F/2)=dt
Multipliying the above both sides by 1/2
d(F/2)/(3/4 – F/2) = dt/2
then integrating the both sides
ln(3/4 – F/2) = t/2 + C
where C is the integrating constant. Inserting the initial values of t=0 and F=3, we find
ln(3/4 – 3/2) = 0 + C
Hence, we can write
ln(3/4 – F/2) + ln(3/4 – 3/2) = t/2
or
ln(3/4 – F/2) – ln(3/4 – 3/2) = t/2
Collecting the left side gives
ln[(3/4 – F/2)/(3/4 – 3/2)]= t/2
Next, we write as following
[(3/4 – F/2)/(3/4 – 3/2)]= exp(t/2)
[(1 4/3* F/2)/(1 4/3* 3/2)]= exp(t/2)
1 – 2F/3 = (1 2)exp(t/2)
1 – 2F/3 = exp(t/2)
2F/3 =1 + exp(t/2)
Finally, we obtain the solution of 4dF/dt=32F to be equal to the result created by using Rohedi’s Formula.
Thx mr.Rohedi your formula is veryvery smart.
Comment by Denaya Lesa  January 6, 2009
Oh yeah?
Solving the linear first ODE using Rohedi’s Formula is more simplier.
Comment by Ahmed Adib  January 6, 2009
Actually there is interesting discussion on http://eqworld.ipmnet.ru/forum/viewforum.php?f=2 It was a new concept for solving ODE, creating euler number without using argand diagram, and so many. I visit on his website rohedi.com few months ago. The most interesting is that rohedi formula can eliminate Bessel, Airy and some other function.
Comment by Andrea  January 7, 2009
@Andrea
What you mean? My difficulties for registration to rohedi.com to download the papers and articles.
Comment by Richard  January 7, 2009
I have no ideas about this topic except salute for Mr. Rohedi who has a smart formula for the linear first ODE. I believe by using the Rohedi`s formula the exploring phenomenon in nature more easier. Good luck Mr. Rohedi…
Comment by Shofiyullah Mz  January 7, 2009
Good luck metrostateatheists.wordpress.com can be visited from this address
http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34&st=0&sk=t&sd=a&start=30
Comment by jacky  January 7, 2009
Yes,I also read at
http://autarkaw.wordpress.com/2008/07/21/timeofdeathaclassicodeproblem/
Rohedi’s Formula for Bernoulli differential equation posted by Denaya
dy/dx + Py = Qy^n, for n is not equal 1
that is of form :
y(x)=[{y0^(1n) – Q/P}exp{(1n)p(xx0)} + Q/P ]^{1/(1n)}
But I don’t know it’s applications.
Comment by jacob  January 7, 2009
@Richard, maybe the server is busy, you can try it later. About eliminate some function(Bessel, Airy etc) you can ask to Mr Rohedi directly, I am sure he will give explanation.
Comment by Andrea  January 8, 2009
Ohh I’ve just known that there are problems with the registration to my website. But I think what delivered by Andrea is the cause.
@ Richard
Generally the solution of homogeneous linear second order ODE are in special functions, such as Airy function, Bessel function, Kummer function, Hypergeometric function etc.
The special functions are generated by using Frobenius method that involved the power series expansion. The alternative way that has been commonly performed in solving the second order ODE that is by transforming into nonlinear first order socalled Ricatti’DE. But the main problem, the general solution of Ricatti’s DE until now has not been found. With an appropriate way, I believe the SMT can solve the general Ricatti’s DE. If this is correct, of course we need not all of the special functions again.
Comment by rohedi  January 8, 2009
Creating solution of the linear first order ODE and the Bernoulli’s DE have been commonly teached in Faculty. But, creating solution by SMT, oh my GOD, it’s smart solver for the first order ODE. My question, whether the solving procedure by this Stable Modulation Technique is chatagorized as bad math?
Comment by Han  January 9, 2009
@Han,
Excelent’s question, how about you all?
Comment by Denaya Lesa  January 9, 2009
@ Denaya Lesa,
I don’t know about classifying the SMT as bad or good math. Mr.Yono, my colleague ever discussed about this on this address
http://math152.wordpress.com/2008/10/08/mathandtheoriginsoflife/#comment218
but he tells me that until now they have not yet given a respon.
Be patient miss Denaya, now I am preparing special paper of SMT’s application in solving the general Ricatti’s DE that will be submitted to appropriate Mathematics Journal.
Comment by rohedi  January 10, 2009
Apologize, my above reply holds to mr.Han also.
Regards.
Comment by rohedi  January 10, 2009
Okay, finally I inform you all that there are sixteen classes of the Ricatti’s DE that can be solved exactly easily as explained at math book of Polyanin et al. But, until know the general solution of the Ricatti’s DE dy/dx=p(x)y^2+q(x)y+r(x) has not been found. I will report that the SMT can solve the nonlinear first order ODE easily.
Comment by rohedi  January 10, 2009
Hi all,
This address http://randualamsyah.wordpress.com also links to this metrostateatheis.wordpress.com.
Please visit to one of the blog in Indonesia. Thx.
Comment by Denaya Lesa  January 11, 2009
All,
I feel overconfidance to post Rohedi’s Formula for Bernoulli Differential Equation (BDE) in math web, for example at this address
http://collegemath.wordpress.com/2008/05/31/example/#comment15
Comment by Denaya Lesa  January 12, 2009
Hay hay hay,
I ever read on
http://unapologetic.wordpress.com/2008/10/13/sineandcosine/
that are another example of differential equation that related directly with calculus (here integral). Please read comment Mr.Rohedi (http://rohedi.com).
f”+f=0
I think you have done big blunder, huahuahau…
Comment by Denaya Lesa  January 14, 2009
Sorry, the two last line at my comment above must be deleted, he he he …I copied it from my sister’s comment Nadya Fermega at
http://trevorpythag.wordpress.com/2009/01/08/differentiateinversecosproof/#comment185
Comment by Denaya Lesa  January 14, 2009
Dear All,
I am interested to the topic of stochastic phenomenon especially in order to explore the corrosion mechanism in nano structures to support my research group (at my Dept of Physics, Sepuluh Nopomber Institute of Technologi (ITS) Surabaya, Indonesia) in developing Nano Technology at my country.
You know one of stochastic model that has been commonly used for the purpose is in the following nonlinear first order differential equation driven by white noise,
dy/dt+py=Qy^3 + Acos(2Пft)+δ(tt’)
where δ(tt’) is the white noise mentioned.
Of course imposible to obtain analytic solution of the above model. But I believe analytic solution of ordinary part of the DE will help in creating accurate solution of the stochastic model.
Recently I have been developing a new method socalled SMT (stable modulation technique) for solving first order ordinary differential equation (ODE) based applying a new modulation scheme that I introduced as stable modulation in which solution of linear solution part of the ODE is substituted into amplitude of the nonlinear solution part. Here, solution of the nonlinear part must be written in a modulation function AF(A) that it’s amplitude A also including at the phase function. A. The SMT succesfully in solving the general Bernoulli Differential Equation (BDE)
dy/dx+p(x)y=Q(x)y^n, for n≠1
or another ODE that can be transformed into the BDE. As I informed in this forum previously, the utilize of SMT has been posted at http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34&start=20.
Now, I invite you to collaborate with me especially to develop the SMT for solving the general of inhomogenous of BDE:
dy/dx+p(x)y=Q(x)y^n + f(x), for n≠1
where f(x) is arbitrary force function.
Okay, I wait you all who are interested to my purpose. You can contact me via email at aliyunus@rohedi.com, or by leaving some comments at my website http://rohedi.com and/or at my wordpress http://rohedi.wordpress.com.
Thank you very much for your attention.
My best Regards,
AFASMT/Rohedi.
Comment by afa  January 16, 2009
All,
I read at
http://eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34&start=30
there is an apportunity to collaborate in developing the SMT to solve the key mathematics model of developing Nano Technology,
dy/dx+p(x)y=Q(x)y^n + f(x) , n≠1
Please visit to the above address.
Comment by Denaya Lesa  January 16, 2009
Hi All,
This blog of contra religion is very inspiring for me. But before giving further explanation about this, please visit and leave some comments for my posts at this link.Thx.
http://trevorpythag.wordpress.com/2009/01/28/derivequadraticformula/
Comment by Denaya Lesa  February 6, 2009
After visiting to
http://blueollie.wordpress.com/2009/02/05/dailykosthepresidentstrikesback/#comment31157
and visit here, OMG, hehehe….as I told previously that this blog is very inspiring hence so interesting to be visited.
Here I find a new definition “Light=God”, but this is not important for me whether hehe… the keyword is wrong or not. I am more interested to the subject of light, in which according to our teacher it contains of electric and magnetic fields, where in free space both of fields propagate perpendicular of each others, weren’t there.
Next, I invite you all here visit to
http://trevorpythag.wordpress.com/2009/01/22/equationsofmotion/
On the above math blog, of course guided by my father Mr.Rohedi, I will discuss a simpliest way of solving mathematics model for harmonically motion, that is of form :
s”(t) + [(2*pi*f)^2]s(t) = 0
where s, f, and t are displacement, frequency, and time elapsed respectively.
What is the primary message from the nice model?
We are all together like Adam and Eve must life and living in this world harmonically of each others. How about you all?
Happy with you,
Denaya Lesa.
Comment by Denaya Lesa  February 6, 2009
Hello Denaya,
Light is indeed a very interesting and mind boggling phenomena, and one which has nothing to do with and deity.
– Chalmer
Comment by Metro State Atheists  February 7, 2009
@Metro State Atheists and All
All rights, when we hear the light word, surely that imagined on our brains is the various of colors red, orange, yellow, green, blue, ultra violet, etc that related with the frequency or wavelength of electromagnetics that arrived to our eyes. According to my physics teacher the electromagnetics spectrum eventually were associated with Planck’s Formula of blackbody radiation’s spectral density that is of form
U(T) = ћω / [exp(ћω /kT) – 1]
Okay, before Denaya relates Planck’s Formula with relegion aspect,there is challenge for you. Anyone here that can prove the equality of the above common formula form with the new form of Planck’s Formula below
U(T) = 0.5 ћω [ tanh{0.5ћω /kT – iП/2} – 1]
that released by my father Mr.Rohedi? Okay, Denaya is still be patient to wait your answer.
Yours sincerely,
Denaya Lesa.
Comment by Denaya Lesa  February 9, 2009
Dear @MSA,
Recently your article about differential equations how they relate to calculus becomes top clicks on:
http://blueollie.wordpress.com/2009/03/15/piday2009sortof/#comment33251
On the Blueollie’s website I informed that on the following address:
http://eqworld.ipmnet.ru/forum/viewtopic.php?f=3&t=148,
there are a post about a New Exact Formula for Pi Number that it’s general form in the sum of two arcsin functions. Please visit to the link, maybe useful for preparing the next years Pi Day.
Oh yeah, there is a special sallom from my daughter Denaya Lesa. Now she focuses to prepare her final exam.
Thanks you,
Best Regards,
Rohedi.
Comment by Rohedi  April 12, 2009
great work helped me to write an assignment on solution of differential equations
Comment by zeeshan  June 4, 2009
You’re welcome!
Comment by Metro State Atheists  June 4, 2009
I am a college student of civil engeenering that now in preparing my final project related to damped oscillation. I need your information is there shortcut way for solving differential equation of second Newton law for small damped oscillation m(d^2)x/dt^2=bdx/dtcx of object with m mass undergoes a small damped, while both of b and c are constants. Thanks four your attention.
Comment by Rokib  June 20, 2009