Solving a Differential Equation
Date: 6/5/96 at 21:32:37 From: Anonymous Subject: Elementary Solution? Dr. Math: When I solved the first order differential equation x(y')+xy = 1 (*) by dividing by x, then multiplying by the integrating factor, and obtained a solution which required me to solve an integral looking somewhat like this S ((e^u)/u) du the integral resisted every attack on my part. I was able to iteratively apply a modified version of integration by parts to the integral above, obtaining the following series: 00 (<--infinity) .----- \ (k-1)! (e^u) * . ------- (***) / u^k .----- k = 1 Intuitively, I would say the differential equation (*) has no solution expressable in terms of elementary functions. How would I go about finding out about this? In other words, is there a way to *prove* that (***) above does not belong to the set of elementary functions? I would like to note the resemblance of the summation part in (***) to that of the Taylor series expansion of e^z 00 .------ \ z^(k-1) e^z = . -------- / (k-1)! .------ k = 1 So, is there a manner to solve the diff.eq. (*) such that an elementary form of (***) will be obtained? P.S. In case you want to know, the version of the integration by parts I used is: S (1/a) db = a/b + S (a/b^2) da which can be proven quite easily.
Date: 6/6/96 at 6:13:59 From: Doctor Anthony Subject: Re: Elementary Solution? There is in fact a solution in series to the differential equation x(dy/dx) + xy = 1 dy/dx + y = 1/x e^(x)(dy/dx) + e^(x)*y = e^(x)/x and integrating we get y*e^x = INT[e^(x)dx/x] = ln|x| + x/1! + x^2/(2.2!) + x^3/(3.3!) + x^4/(4.4!) + ... -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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