Differential Equations Ordinary and Partial Differential Equations Math Forum

 September 19th, 2019, 11:44 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math Non-order DE Solve equation : $\displaystyle y''=y+e^{2x}$ . September 19th, 2019, 01:14 PM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 380 Thanks: 205 $y = \Sigma c_n e^{\lambda_n x}$ $\lambda = 2$ or $\lambda^2 - 1 = 0$ $\rightarrow y=\frac{1}{3} e^{2x} + c_1 e^x + c_2 e^{-x}$ $c_1, ~c_2$ from initial/boundary conditions Thanks from topsquark and idontknow September 19th, 2019, 08:28 PM #3 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 $e^{-x}y'' - e^{-x}y = e^x \\ e^{-x}y' + e^{-x}y = e^x + 2\text{A} \\ e^xy' + e^xy = e^{3x} + 2\text{A}e^{2x} \\ e^xy = \frac13e^{3x} + \text{A}e^{2x} + \text{B} \\ y = \frac13e^{2x} + \text{A}e^x + \text{B}e^{-x}$ Thanks from topsquark, v8archie, idontknow and 1 others September 21st, 2019, 05:48 AM #4 Senior Member   Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math Looks similiar to Wronskian . $\displaystyle dW(y,e^{-x} )=e^{x}dx$. Thanks from topsquark September 21st, 2019, 05:28 PM   #5
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Quote:
 Originally Posted by skipjack $e^{-x}y'' - e^{-x}y = e^x \\ e^{-x}y' + e^{-x}y = e^x + 2\text{A} \\ e^xy' + e^xy = e^{3x} + 2\text{A}e^{2x} \\ e^xy = \frac13e^{3x} + \text{A}e^{2x} + \text{B} \\ y = \frac13e^{2x} + \text{A}e^x + \text{B}e^{-x}$
Very nice.

The more standard approach would be to solve the characteristic polynomial for $y''-y=0$ ($r=\pm1$) and them find a particular solution for the original equation, probably using the method of undetermined coefficients with $y_p=Ae^{2x}$. Tags nonorder 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 Don96 Differential Equations 4 April 22nd, 2016 08:14 PM math2016 Differential Equations 2 October 2nd, 2015 09:04 AM Kappie Abstract Algebra 0 April 22nd, 2012 02:52 PM Grayham1990 Calculus 2 March 30th, 2012 07:24 AM Norm850 Calculus 2 March 7th, 2012 05:08 PM

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