Metro State Atheists

Promoting Science, Reason, and Secular Values

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, science, Uncategorized | , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,


  1. Maybe useful for you all, in the following address
    afasmt informs the discussion of Bernoulli integral related with Bernoulli differential equation that created by a new solver of nonlinear first order ODE by so-called Stable Modulation Technique.

    Comment by rohedi | January 5, 2009

  2. Thank you very much for this additional information.

    Metro State Atheists

    Comment by Metro State Atheists | January 5, 2009

  3. @rohedi

    After visiting to

    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)=[y0-Q/P]exp{-P(x-x0)} + Q/P

    Now, suppose we have a problem 4dF/dt=3-2F, 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


    F(t) = (3/2)[exp{-t/2} + 1]

    Please verify the above solution with the usual way.

    Comment by Nadya Fermega | January 6, 2009

  4. @Nadya Fermega

    An interesting discussion. I am interested to verify the solution of 4dF/dt=3-2F 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


    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=3-2F to be equal to the result created by using Rohedi’s Formula.

    Thx mr.Rohedi your formula is very-very smart.

    Comment by Denaya Lesa | January 6, 2009

  5. Oh yeah?
    Solving the linear first ODE using Rohedi’s Formula is more simplier.

    Comment by Ahmed Adib | January 6, 2009

  6. Actually there is interesting discussion on It was a new concept for solving ODE, creating euler number without using argand diagram, and so many. I visit on his website few months ago. The most interesting is that rohedi formula can eliminate Bessel, Airy and some other function.

    Comment by Andrea | January 7, 2009

  7. @Andrea

    What you mean? My difficulties for registration to to download the papers and articles.

    Comment by Richard | January 7, 2009

  8. 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

  9. Good luck can be visited from this address

    Comment by jacky | January 7, 2009

  10. Yes,I also read at

    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^(1-n) – Q/P}exp{-(1-n)p(x-x0)} + Q/P ]^{1/(1-n)}

    But I don’t know it’s applications.

    Comment by jacob | January 7, 2009

  11. @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

  12. 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 so-called 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

  13. 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

  14. @Han,

    Excelent’s question, how about you all?

    Comment by Denaya Lesa | January 9, 2009

  15. @ 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

    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

  16. Apologize, my above reply holds to mr.Han also.


    Comment by rohedi | January 10, 2009

  17. 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

  18. Hi all,

    This address also links to this
    Please visit to one of the blog in Indonesia. Thx.

    Comment by Denaya Lesa | January 11, 2009

  19. All,

    I feel overconfidance to post Rohedi’s Formula for Bernoulli Differential Equation (BDE) in math web, for example at this address

    Comment by Denaya Lesa | January 12, 2009

  20. Hay hay hay,

    I ever read on

    that are another example of differential equation that related directly with calculus (here integral). Please read comment Mr.Rohedi (


    I think you have done big blunder, huahuahau…

    Comment by Denaya Lesa | January 14, 2009

  21. 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

    Comment by Denaya Lesa | January 14, 2009

  22. 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(t-t’)

    where δ(t-t’) 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 so-called 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

    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, or by leaving some comments at my website and/or at my wordpress

    Thank you very much for your attention.

    My best Regards,


    Comment by afa | January 16, 2009

  23. All,

    I read at

    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

  24. 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.

    Comment by Denaya Lesa | February 6, 2009

  25. After visiting to

    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

    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

  26. 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

  27. @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 black-body 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

  28. Dear @MSA,

    Recently your article about differential equations how they relate to calculus becomes top clicks on:

    On the Blueollie’s website I informed that on the following address:,

    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,

    Comment by Rohedi | April 12, 2009

  29. great work helped me to write an assignment on solution of differential equations

    Comment by zeeshan | June 4, 2009

  30. You’re welcome!

    Comment by Metro State Atheists | June 4, 2009

  31. 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/dt-cx 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

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