Differential Equations Ordinary and Partial Differential Equations Math Forum

April 22nd, 2019, 08:55 PM   #1
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Undetermined coefficients

Hello everyone,
In this equation what will be the answer guess for 12t^-2e^(-3t/2)

I know polynomial guess
Trigonometric guess
Exponential guess
But didn’t come across t^negative power Thanks
Attached Images 5CFB888B-F2BF-4AD2-8087-350724298A17.jpg (11.7 KB, 18 views) April 23rd, 2019, 05:21 AM #2 Senior Member   Joined: Dec 2015 From: Earth Posts: 832 Thanks: 113 Math Focus: Elementary Math This equation can be solved with numerical methods.There may be no elementary solutions. April 23rd, 2019, 12:04 PM #3 Senior Member   Joined: Sep 2015 From: USA Posts: 2,644 Thanks: 1476 we have a homogeneous solution $c_1 e^{-3t/2} + c_2 t e^{-3t/2}$ clearly the particular solution won't be exactly one of these forms so let's try $p(t) = c_3 e^{-3t/2}f_3(t) + c_4 t e^{-3t/2} f_4(t)$ and run it through the diff eq and see what we see. $4p''(t)+12p'(t)+9p(t) = 4 e^{-\frac{3 t}{2}} \left(c_3 \text{f3}''(t)+c_4 \left(t \text{f4}''(t)+2 \text{f4}'(t)\right)\right)$ It appears we can simplify things by setting $c_4=0$ to obtain $4c_3e^{-3t/2}f3''(t) = 12t^{-2}e^{-3t/2}$ This gets you a differential equation in $f(t)$ which is easily solved by repeated integration. I'm going to let you grind through it all but it appears that a solution that works is $p(t) =-3 e^{-3t/2}(c t+\ln (t)-3),~c \in \mathbb{R}$ Thanks from idontknow and Leonardox April 23rd, 2019, 04:26 PM #4 Global Moderator   Joined: Dec 2006 Posts: 21,114 Thanks: 2329 $(4y'' + 12y' + 9y)e^{3t/2} = 12t^{-2}$ Integrating and dividing by 4 gives $(y' + (3/2)y)e^{3t/2} = -3/t + \text{c}_1$. Integrating again gives $ye^{3t/2} = -3\ln|t| + \text{c}_1t + \text{c}_2$. Thanks from idontknow and Leonardox April 24th, 2019, 06:55 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra I'm going to suggest that, if you can't think of the template to use for the Method of Undetermined Coefficients, you should be using the Method of Variation of Parameters. Thanks from idontknow April 24th, 2019, 08:21 AM   #6
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Math Focus: Elementary Math
Quote:
 Originally Posted by idontknow This equation can be solved with numerical methods.There may be no elementary solutions.
My mistake on reading the attachment. Tags coefficients, undetermined 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 aliaja Applied Math 1 October 19th, 2013 09:06 AM BinaryReader Calculus 1 May 13th, 2012 08:17 PM mbradar2 Differential Equations 2 October 26th, 2010 02:10 PM mbradar2 Calculus 1 October 21st, 2010 07:46 AM jc_m15 Calculus 3 April 3rd, 2008 12:07 PM

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