Need Help on a TOUGH Calculus question

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thread starter silkym39 Feb 21 2010 + PM | QUOTE | PERMALINK | REPORT

Can anyone solve this step by step for me? I have tried to solve it, but it seems impossible: Show that the function y= (fx) is a a solution of the differential equation. y = e^x[3cos(2x) - 4sin(2x)] y" - 2y' + 5y = 0 This question is from the book Calculus of a Single Variable , Section 5.4 #64

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Senyain + PM | QUOTE | PERMALINK | REPORT

It's not really all that bad, actually. First derivative (product rule): e^x (3 cos 2x - 4 sin 2x) + e^x (-6 sin 2x - 8 cos 2x) = e^x (3 cos 2x - 8 cos 2x - 4 sin 2x - 6 sin 2x) = e^x (-5 cos 2x - 10 sin 2x) = -5 e^x (cos 2x + 2 sin 2x) Second derivative (product rule): -5 e^x (cos 2x + 2 sin 2x) - 5 e^x (-2 sin 2x + 4 cos 2x) = -5 e^x (cos 2x + 4 cos 2x + 2 sin 2x - 2 sin 2x) = -5 e^x (5 cos 2x) = -25 e^x cos 2x Okay, now to prove it's a solution for y'' - 2y' + 5y = 0 is just a matter of plugging those in. This is going to involve mass cancellations. Check out how neatly this breaks down: -25 (e^x cos 2x) - 2 (-5 e^x [cos 2x + 2 sin 2x]) + 5 (e^x [3 cos 2x - 4 sin 2x]) = -25 e^x cos 2x + 10 e^x cos 2x + 20 e^x sin 2x + 15 e^x cos 2x - 20 e^x sin 2x It might look messy but notice EVERYTHING is either e^x cos 2x or e^x sin 2x. You remember your distributive property, right? e^x cos 2x (-25 + 10 + 15) + e^x sin 2x (20 - 20) = 0 + 0 = 0.

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Oogendune 02/21/10 + PM | QUOTE | PERMALINK | REPORT

Don't forget that the derivative of e^x is e^x .. and the derivative of cosx is -sinx and the derivative of sinx is cosx

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Logic 02/21/10 + PM | QUOTE | PERMALINK | REPORT

If you wanted f'(x), it would be e^(x(3cos(2x) - 4sin(2x)) multipled by (x(-6sin(2x) - 8cos(2x) + 3cos(2x)- 4sin(2x)) So e^(x(3cos(2x) -4sin(2x)) x (-6xsin(2x) - 8xcos(2x) + 3cos(2x) - 4sin(2x)) Enjoy :D

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Retro 02/21/10 + PM | QUOTE | PERMALINK | REPORT

y' = e^x[-6sin(2x) - 8cos(2x)] + e^x[3cos(2x)-4sin(2x)] = e^x[-10sin(2x) - 5cos(2x)] y'' = e^x[-20cos(2x) + 10sin(2x)] + e^x[-10sin(2x) - 5cos(2x)] = e^x[-25cos(2x)] e^x[-25cos(2x)] - 2e^x[-10sin(2x) - 5cos(2x)] + 5e^x[3cos(2x) - 4sin(2x)] = 0 O_O"

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cb000 + PM | QUOTE | PERMALINK | REPORT

[quote] Logic : Integrals are so easy. When I saw this first question, I wanted to do separation of variables, but he only had one variable. The only difficulty of this problem is that your e^(A) is a product rule, which you have to chain rule out. [/quote] Easy? Have you tried integrating the square root of sin(x)? It gets messy, especially when the elliptic integrals come in. Or maybe even the square root of tan(x), though that's not as messy. OT: Looks like everyone's essentially doing the same thing. Get the derivatives, and plug them in.

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Divination 02/21/10 + PM | QUOTE | PERMALINK | REPORT

ya umm the real question is will solving for y have any use in the real world workforce of today?

BePersian 02/21/10 + PM | QUOTE | PERMALINK | REPORT

[quote] iODST : yes uhm about that im in 7th grade in algerbra 1 and is calc actually that hard cause ive been getting C's in algerbra [/quote] LOL you should prolly stop at pre-calc then

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metaghost3 02/21/10 + PM | QUOTE | PERMALINK | REPORT

Aww crap don't tell me i screwed up somewhere

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Senyain 02/21/10 + PM | QUOTE | PERMALINK | REPORT

[quote] Divination : ya umm the real question is will solving for y have any use in the real world workforce of today? [/quote] Let's see... by my last count, calculus is used in... all sophisticated forms of engineering, architecture, computer science (including programming the computers that can solve these derivatives quickly), physics, sports analysis, anything involving a rate of change, radio wavelengths, actuarial/financial work... need me to keep going?

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metaghost3 02/21/10 + PM | QUOTE | PERMALINK | REPORT

Oh forget i didn't even pay attention to the question :P

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Divination 02/21/10 + PM | QUOTE | PERMALINK | REPORT

[quote] Senyain : Let's see... by my last count, calculus is used in... all sophisticated forms of engineering, architecture, computer science (including programming the computers that can solve these derivatives quickly), physics, sports analysis, anything involving a rate of change, radio wavelengths, actuarial/financial work... need me to keep going? [/quote] the question is: do u intend on using calculus in your life to help you do the aforementioned things above?

jameschen0 02/21/10 + PM | QUOTE | PERMALINK | REPORT

AoPS works wonders

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6ReDevil6 + PM | QUOTE | PERMALINK | REPORT

Im suprised u didnt just factorize the e^x senyain. And for people that ignore the utility of math, please open your minds and think critically... Common look at your computer!

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Retro 02/21/10 + PM | QUOTE | PERMALINK | REPORT

[quote] Divination : the question is: do u intend on using calculus in your life to help you do the aforementioned things above? [/quote] Simply put, yes. I think you should just continue to respond to obvious calculus questions with "Trig is easy. Use a calculator." Lmao.

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Senyain 02/21/10 + PM | QUOTE | PERMALINK | REPORT

[quote] Divination : the question is: do u intend on using calculus in your life to help you do the aforementioned things above? [/quote] Seeing as I student-teach math right now and will be teaching it full-time in less than a year, I can only respond with a resounding YES. Furthermore, before I decided I'd rather make a difference to people than be a mercenary millionaire, I was thinking of becoming an actuary, another field in which I would have used this info.

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