Math question: basic, basic differential equations

[ZenMaster]

I could probably go to a proper "Math Tutor" website but I'm lazy.

I've recently gone WAY back to the beginning of my serious education in math. I eventually want to teach myself General Relativity and Quantum Physics, but even my earliest "serious" math skills have become hideously rusty.

In particular, the following question from an intro chapter to my old textbook on differential equations has got me stumped:

http://4.bp.blogspot.com/_7qfwpjecuw8/THXYSo4ypNI/AAAAAAAAABk/5j8ouZFH0aY/s1600/Problem14_4b.bmp

The goal is to determine if the given function y is a solution of the differential equation. The second term I can handle and I remember integration by parts, but it's the inside of the integral that throws me. I've forgotten how to integrate e to a non-trivial function (i.e. anything but s).

Can anyone help?

[Cartesiantheater]

I could probably go to a proper "Math Tutor" website but I'm lazy.

I've recently gone WAY back to the beginning of my serious education in math. I eventually want to teach myself General Relativity and Quantum Physics, but even my earliest "serious" math skills have become hideously rusty.

In particular, the following question from an intro chapter to my old textbook on differential equations has got me stumped:

http://4.bp.blogspot.com/_7qfwpjecuw8/THXYSo4ypNI/AAAAAAAAABk/5j8ouZFH0aY/s1600/Problem14_4b.bmp

The goal is to determine if the given function y is a solution of the differential equation. The second term I can handle and I remember integration by parts, but it's the inside of the integral that throws me. I've forgotten how to integrate e to a non-trivial function (i.e. anything but s).

Can anyone help?


I am skeptical that this problem came from "an intro chapter." (unless it came from a graduate textbook). This problem is no picnic. You are not going to be able to find a nice anti-derivative to that integral. It cannot be expressed as simple elementary functions. This integral is a constant times the error function. (unless I missed something important, which is possible- first day of classes. I'm quite busy)

http://en.wikipedia.org/wiki/Error_function


The way you will have to deal with this, I believe, is to express it as an infinite series (you can find this on the wiki page, I believe), add your two functions (y'' and y), and see if that series looks like sec(t) -(obviously if you do it this way you'll need to expand secant (http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/mathbas/sec.htm) as well).





As for everything else, here is what I'd do:




Take the first and second derivatives of y, then sum the second derivative with the function. If it IS a solution, then that sum will be sec t, over your interval.

The hard part is the derivatives.


Here is how I would do it:

(1) Use the fundamental theorem of calculus for y' and y'' to deal with the integral for the y'' part of your equation.

http://www.mathmistakes.info/facts/CalculusFacts/learn/doi/doi.html



(2) Take the second derivative.

(3) Add the second derivative to the original function (I would do this with the integral already integrated so that you have nothing but a function of t). That is, y'' + y.

(7) if y'' + y = sec t, you win. However, keep in mind that you might have to use Euler's formula to rewrite all these exponential functions as trig functions.


But again, the integral of 0 to x of e^-x^2 dx cannot be expressed in terms of elementary functions. You will have to express it as an infinite series, which you can look up.







One thing you might consider trying instead of that- and I haven't tried it nor have I looked to hard at it, since I have class in about 20 minutes, is this:



You're trying to see if y is a solution to y'' + y = sec(t)

Well, if you ASSUME that it is, then I imagine you can write y = sec(t) - y''

You can easily get y'' using the fundamental theorem (and all the product rule usage you're going to have to do).

http://img829.imageshack.us/img829/9254/fundamentaltheoremofcal.png (http://img829.imageshack.us/i/fundamentaltheoremofcal.png/)

Uploaded with ImageShack.us (http://imageshack.us)




Then examine the expansions and see if it matches (you may not have to do this, though- again I haven't looked to hard. I might tomorrow when I get the internet in my apartment).








One final thing: e^f(t) * integral of e^f(s) from 0 to t looks very much like what I did last year when studying Laplace transforms and things of that nature. You MIGHT want to look into that. I don't have my text or notes on me so I can't look it up, but that might be something you want to look at.

[Greatest I am]

Test test test

[ZenMaster]

Yup, CT had it right re: the Error Function. I'm not certain why that advanced question was in the beginning of the textbook either.