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guess

  • 2 years ago

find the general solution y``-5y`-6y=cos x

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  1. scarydoor
    • 2 years ago
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    this is a 2nd order linear differential equation. There is a really easy method for solving these things. Do you know of it? You should have been told it in class. Otherwise you can search google and it'll tell you the method.

  2. Aperogalics
    • 2 years ago
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    @guess it is a second order equation it as two parts one general and particular and their sum is the complete solution so for general put y''-5y'-6y=0 and replace y^n by D^n and solve quadratic...............try it:)

  3. Aperogalics
    • 2 years ago
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    @RolyPoly let him solve:)

  4. RolyPoly
    • 2 years ago
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    First, find the complementary solution by setting the left side equal to 0. y``-5y`-6y=0 <- homogeneous solution, so, λ^2 - 5λ -6 =0 Solve λ \[y_c = c_1e^{\lambda _1 x} + c_2e^{\lambda _2 x}\] Then, find out the particular solution, try \(y_p = Asinx+Bcosx\) (No guarantee that it works though). Differentiate it and solve the A and B. Then, you get \(y_p\) \[y = y_c +y_p\] @Aperogalics I never intend to give out the answer..

  5. guess
    • 2 years ago
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    y(h)=c1e^(-x)+c2e^(6x) g(x)=cos x y(P)=a cosx+b sin x >> I stood at the 5A-7B=0 -(5B+7A)=0 Then ??

  6. guess
    • 2 years ago
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    @RolyPoly @Aperogalics @scarydoor thank you but i need some help

  7. RolyPoly
    • 2 years ago
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    Hmm... Do you mind showing your work for the particular solution?

  8. scarydoor
    • 2 years ago
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    It's called 'method of undetermined coefficients'. If you're going to be doing differential equations a fair bit, then I recommend this book: http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_1?ie=UTF8&qid=1354192173&sr=8-1&keywords=ordinary+differential+equations It's about the best book you can get for it.

  9. scarydoor
    • 2 years ago
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    Do you know how it works? The method to find a particular solution? That seems to be what you're stuck on. This is the method to use: http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients

  10. guess
    • 2 years ago
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    Of course @RolyPoly y(P)=a cosx+b sin x y`=-A sin x+b cos x y``=a cos x-b sin x y``=-y -y-5y`(-a sin x+b cos x)-6y =cos x 5asinx-5b cos x-7acosx -7b sin x=cos x (5a-7b)sinx-(5b+7a)cos x=cosx 5A-7B=0 -(5B+7A)=0

  11. guess
    • 2 years ago
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    @scarydoor thanks

  12. RolyPoly
    • 2 years ago
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    y`=-A sin x+b cos x Isn't y'' = -Acosx - Bsinx?

  13. RolyPoly
    • 2 years ago
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    But that doesn't affect your conclusion that y'' = -y y(P)=a cosx+b sin x y`=-A sin x+b cos x y``=-a cos x-b sin x = -y So, -y-5y`-6y =cos x -7 y - 5y' =cos x (5a-7b)sinx-(5b+7a)cos x=cosx 5a-7b=0 -(5b+7a)=0 (it should be -(5b+7a) =1?!)

  14. RolyPoly
    • 2 years ago
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    When you get the two equations: 5a-7b =0 and -(5b+7a) =1 , you can solve a and b and you'll get the particular solution.

  15. guess
    • 2 years ago
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    @RolyPoly no maybe -5b-7a

  16. RolyPoly
    • 2 years ago
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    (5a-7b)sinx-(5b+7a)cos x=cosx Compare the terms on the left and right, there is no sinx term, so the coefficient of sinx is 0 Hence you get the first equation 5a-7b =0 there is a cosx term, with the coefficient 1, so the coefficient of cosx term on the left is equal to 1. Hence we should get -(5b+7a) = 1 as the second equation. Got it?

  17. guess
    • 2 years ago
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    @RolyPoly yes thank you so we get a=7/74 b=5/74 ?

  18. RolyPoly
    • 2 years ago
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    Not quite.

  19. RolyPoly
    • 2 years ago
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    You've missed a negative sign for both a and b...

  20. guess
    • 2 years ago
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    @RolyPoly thank you

  21. RolyPoly
    • 2 years ago
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    Welcome :)

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