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April 22nd, 2019, 08:55 PM  #1 
Senior Member Joined: Apr 2017 From: New York Posts: 165 Thanks: 6  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 didnâ€™t come across t^negative power Thanks 
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 $p(t) =3 e^{3t/2}(c t+\ln (t)3),~c \in \mathbb{R}$ 
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 $ye^{3t/2} = 3\lnt + \text{c}_1t + \text{c}_2$. 
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.

April 24th, 2019, 08:21 AM  #6 
Senior Member Joined: Dec 2015 From: Earth Posts: 832 Thanks: 113 Math Focus: Elementary Math  

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