My Math Forum non-homogeneous recurrence problem

 Applied Math Applied Math Forum

 May 20th, 2019, 09:39 AM #1 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 non-homogeneous recurrence problem Hello everyone, I have this problem $a_n + a_{n-1} + 6a_{n-2} = 5n(-1)^n + 2^n$ This is the given solution: $\displaystyle a_n = (\sqrt{6})^n(C_1\cdot \sin\phi n + C_2\cdot \cos\phi n) + \frac{1}{3} \cdot 2^n + \left(\frac{5}{6}\cdot n + \frac{55}{36}\right)(-1)^n$ I used the discriminant to find the zeros of characteristic polynomial but then I got this: $\displaystyle x_{1,2} = -1 \pm \frac{\sqrt{1-24}}{2}$ and have no idea how to continue. Can someone please help me solve this step by step? Thanks in advance. Last edited by skipjack; May 21st, 2019 at 06:38 AM.
 May 20th, 2019, 11:42 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra That expression for $x_{1,2}$ is going to form the basis of your complementary solution (to the homogeneous equation), the $(\sqrt6)^n\big(c_1\sin (n\phi) + c_2 \cos (n\phi)\big)$, I guess. (The $\sqrt6$ is the norm of the complex roots). The rest of it you can get by the Method of Undetermined Coefficients using $A2^n + (Bn+C)(-1)^n$ as your trial solution. I'm not convinced by your values for $x_{1,2}$. Thanks from topsquark Last edited by v8archie; May 20th, 2019 at 11:51 AM.
 May 20th, 2019, 12:00 PM #3 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 Well the determinant's formula is $\displaystyle x_{1,2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ if I'm correct (btw, I did a mistake in the first post) regarding the determinant, where 1 should be in the numerator too. However still, plugging in the discriminant I get the expression with negative square root, is this ok? and if not how it is then? Last edited by skipjack; May 21st, 2019 at 06:47 AM.
May 20th, 2019, 12:14 PM   #4
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,690
Thanks: 2669

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by SuperNova1250 btw, I did a mistake in the first post) regarding the determinant, where 1 should be in the numerator too. However still, plugging in the discriminant I get the expression with negative square root, is this ok?
That's the error I spotted. Your expression with the negative square root is OK. Since $|x_1| = |x_2| = \sqrt{6}$, can you make a guess as to what $\phi$ might be? If you are struggling with that, consider De Moivre's Theorem $$(\cos \theta + i\sin \theta)^n = \cos n\theta + i\sin n\theta$$

Last edited by skipjack; May 21st, 2019 at 07:20 AM.

 May 20th, 2019, 12:21 PM #5 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 Wait, wait, you used the absolute value on $x_1$ and $x_2$, and got square root of 6. Can you please explain this part in detail, as the homogeneous part is the one I am struggling with? I haven't gotten yet to the particular solution. Last edited by skipjack; May 21st, 2019 at 07:21 AM.
 May 20th, 2019, 12:28 PM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra $x_1$ and $x_2$ are complex numbers $a_1 + ib_1$ and $a_2 + ib_2$. What are the absolute values (or modulus, or norm) of these? If you haven't yet met complex numbers, I don't know why you have this problem to solve.
 May 20th, 2019, 12:33 PM #7 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 I have, the answer I got with the discriminant was $\displaystyle \frac{-1 \pm \sqrt{1-24}}{2}$ which is $\displaystyle \frac{-1 \pm 23i}{2}$ How to continue from here? Last edited by skipjack; May 21st, 2019 at 06:56 AM.
 May 20th, 2019, 04:18 PM #8 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra That should be $$\frac{1 \pm i\sqrt{23}}{2}=\frac12 \pm i \frac{\sqrt{23}}2$$
 May 20th, 2019, 10:23 PM #9 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 Well.. and how does that become $\displaystyle \sqrt{6}$? Last edited by skipjack; May 21st, 2019 at 06:58 AM.
 May 21st, 2019, 03:34 AM #10 Member   Joined: Jan 2016 From: / Posts: 43 Thanks: 1 Come on, can anybody solve this step by step? I am really struggling with this problem. Thanks in advance. Last edited by skipjack; May 21st, 2019 at 07:04 AM.

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post reza767 Number Theory 0 April 10th, 2015 02:42 PM BonaviaFx Differential Equations 1 March 29th, 2015 08:06 AM LouArnold Algebra 0 September 24th, 2012 10:37 PM arron1990 Calculus 5 February 21st, 2012 07:22 AM westaf Differential Equations 3 April 7th, 2011 09:29 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top