Differential Equations Ordinary and Partial Differential Equations Math Forum

 April 3rd, 2018, 07:05 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 536 Thanks: 81 Which series can solve ? How to get the solution... $\displaystyle xy'=e^x$ April 3rd, 2018, 09:51 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 That equation can be written as $\displaystyle y'= \frac{e^x}{x}$. The MacLaurin series for $\displaystyle e^x= 1+ x+ \frac{1}{2}x^2+ \frac{1}{6}x^3+ \cdot\cdot\cdot+ \frac{1}{n!}x^n+ \cdot\cdot\cdot$ and so $\displaystyle \frac{e^x}{x}= \frac{1}{x}+ 1+ \frac{1}{2}x+ \frac{1}{6}x^2+ \cdot\cdot\cdot+ \frac{1}{n!}x^{n-1}+ \cdot\cdot\cdot$. Find y by integrating that term by term. Thanks from topsquark April 3rd, 2018, 01:57 PM #3 Math Team   Joined: May 2013 From: The Astral plane Posts: 2,201 Thanks: 899 Math Focus: Wibbly wobbly timey-wimey stuff. Technical point: Since we have to approximate using a power series shouldn't we be using a Taylor expansion instead of a Maclaurin? (Just for the sake of convergence issues. It's practically the same process.) -Dan April 3rd, 2018, 04:13 PM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 If we were given an initial condition of the form y(a)= Y with a not 0, then we would prefer expand $\displaystyle e^x$ in a Taylor's series about a. Thanks from topsquark April 3rd, 2018, 04:40 PM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2643 Math Focus: Mainly analysis and algebra The series given converges everywhere (that it is defined), so there is no need to search for other (more complicated) series. Although, given an initial condition not at zero might make a different series easier to work with. Thanks from topsquark Tags series, solve Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mathdawg Differential Equations 1 December 8th, 2017 06:41 AM life24 Elementary Math 5 March 25th, 2016 09:47 PM uint Calculus 2 October 20th, 2014 03:38 AM elviro Real Analysis 1 January 30th, 2013 03:35 PM mathmusic Calculus 1 March 18th, 2009 11:41 PM

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