First Order Differential Equations By Separation of Variables & A Few Other Problems

JumboJimBo

Hi There,

I am new to this forum.

I was going through some of my old university notes the other day pertaining to calculus and solving first order differential equations via separation of variables, and came across a problem where some of the simplifications I wrote down at the time (20 years ago) from either myself, or the lecturer, just didn't make sense.

The problem probably lies with me; as these days, at work, if I do math, I tend not to do it manually and via software.

So, whilst I remember some of the fundamentals and pretty much know how the basics work; I have forgotten some of the simplifications and shortcuts.

Anyway, the above problem kind of led me to a few other issues too, and I was hoping that by joining this forum someone could assist and/or explain to me where I have gone wrong and what the real explanations for my confusion are.

Would you/anyone mind?

If not I will then upload a document I recently wrote on the matter that hopefully explains the problem(s).

Kind regards,

Jim.

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Forum Staff

JumboJimBo

OK, thanks SJ.

I need to make sure I have re-written it out properly first.

As whilst the math itself is not overly complex some of the simplifications used combined with my non/directly related issues with it probably overcomplicate it; but since I want to discuss them too I better make sure I have laid it all out properly.

Iâ€™ll return with the problem and some questions.

Cheers,

Jim.

JumboJimBo

Actually, before I do that.

If I may, I will ask this question first.

Please bear with me if it seems meaningless, as it will (hopefully) become clearer why I do this once I complete the aforementioned task within my post #3.

Question: what function of $$\displaystyle \ln$$ (natural logarithm) - when derived - equals $$\displaystyle 1/( x^2+ 1)$$?

I am not looking for $$\displaystyle \arctan$$ $$\displaystyle (x)$$

As I am aware that $$\displaystyle d/dx \arctan(x) = 1/( x^2+ 1)$$ and that $$\displaystyle âˆ« 1/( x^2 + 1)\,dx$$ = $$\displaystyle \arctan (x)$$

I am looking for an antiderivative of $$\displaystyle 1/(x^2+ 1)$$ that is itself:

A) Not a function of $$\displaystyle \arctan$$

B) An explicit natural logarithm function and expression; preferably without using substitution methods such as and/or similar to; $$\displaystyle u= x^2+ 1$$

As I know that

• $$\displaystyle âˆ« 1/( u)$$ $$\displaystyle du$$ = $$\displaystyle \ln$$ $$\displaystyle u$$ + $$\displaystyle C$$

• and that $$\displaystyle d/du$$ $$\displaystyle \ln$$ $$\displaystyle u$$ = $$\displaystyle 1/u$$

Cheers,

Jim.

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skipjack

Forum Staff
The answer must be related to arctan($x$) in some way.

$\dfrac{1}{1 + x^2} = \dfrac{i}{2}\left(\dfrac{1}{x + i} - \dfrac{1}{x - i}\right)$, where $i = \sqrt{-1}$.

Integrating gives $\dfrac{i}{2}(\ln(x + i) - \ln(x - i))$ + C, where C is a constant.

1 person

JumboJimBo

Thanks for that SJ.

I was reasonably sure I was not going to be able to get away from arctan (and/or complex numbers); but just wanted to check with someone that knew more about it than me.

As soon as I get time to write the main problem out Iâ€™ll do it and return with some more questions.

Cheers,

Jim.

JumboJimBo

OK, now I have another problem.

I was just about to upload the equation of question and get you to check the accuracy of it before we continued but . . .

The resulting .pdf is 639KB and the siteâ€™s upload facility wonâ€™t allow me to upload a .pdf file of that size. Additionally, I can't upload the .doc file either, and the calculations I have covered are probably too lengthy to try and neatly/manually type them in the post GUI.

Any suggestions?

Cheers,

Jim.

Joppy

OK, now I have another problem.

I was just about to upload the equation of question and get you to check the accuracy of it before we continued but . . .

The resulting .pdf is 639KB and the siteâ€™s upload facility wonâ€™t allow me to upload a .pdf file of that size. Additionally, I can't upload the .doc file either, and the calculations I have covered are probably too lengthy to try and neatly/manually type them in the post GUI.

Any suggestions?

Cheers,

Jim.

You might have to compress your files.

Also, you can start a new thread when you do.

JumboJimBo

Thanks for that, but it still exceeds the limits when compressed.

Cheers,

Jim.

JumboJimBo

OK . . . .

Finally, I think I have managed to convert the document into an appropriate format for the forum.

Hopefully I can now move forward and get someone that knows more than me to help, answer some questions, and - just maybe - rekindle a love for math I used to have.

So . . .

Before I go ahead with my questions, can someone please check my working/calculations within the attachments?

I went through them again manually over the weekend.

But, since I rarely do this stuff by hand and/or without a software package these days, the chances that I have missed something out and/or gotten something wrong will probably not be insignificant.

Cheers,

Jim.