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Thread: Particular Solution of the differential equation

  1. #1
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    Smile Particular Solution of the differential equation

    Determine :- Particular Solution of the differential equation:

    (12)d^2y/dt2 - 3y = 0

    Boundaries t=0, y=3, dy/dt=0.5


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    Last edited by mrdannydandan; May 27th 2016 at 12:50 PM.
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  2. #2
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    Re: Particular Solution of the differential equation

    what have you tried?

    where are you stuck?
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  3. #3
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    Re: Particular Solution of the differential equation

    Are you taking a course in differential equations? This is a very basic, simple example. Its "characteristic equation" is $\displaystyle 12r^2- 3= 0$ which has roots $\displaystyle r= \pm\frac{1}{2}$ so the "general solution" to the differential equation is $\displaystyle y(t)= Ae^{t/2}+ Be^{-t/2}$ (or equivalently, $\displaystyle y(t)= C cosh(t/2)+ D sinh(t/2)$).

    From the formula $\displaystyle y(t)= Ae^{t/2}+ Be^{-t/2}$, $\displaystyle y'(t)= (A/2)e^{t/2}- (B/2)e^{-t/2}$ so y(0)= A+ B= 3 and y'(0)= A/2- B/2= 1/2.

    From the formula $\displaystyle y(t)= C cosh(t/2)+ D sinh(t/2)$, $\displaystyle y'(t)= (C/2)sinh(t/2)+ (D/2)cosh(t/2)$ so y(0)= C= 3 and y'(0)= D/2= 1/2.
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    Re: Particular Solution of the differential equation

    Quote Originally Posted by HallsofIvy View Post
    Are you taking a course in differential equations? This is a very basic, simple example. Its "characteristic equation" is $\displaystyle 12r^2- 3= 0$ which has roots $\displaystyle r= \pm\frac{1}{2}$ so the "general solution" to the differential equation is $\displaystyle y(t)= Ae^{t/2}+ Be^{-t/2}$ (or equivalently, $\displaystyle y(t)= C cosh(t/2)+ D sinh(t/2)$).

    From the formula $\displaystyle y(t)= Ae^{t/2}+ Be^{-t/2}$, $\displaystyle y'(t)= (A/2)e^{t/2}- (B/2)e^{-t/2}$ so y(0)= A+ B= 3 and y'(0)= A/2- B/2= 1/2.

    From the formula $\displaystyle y(t)= C cosh(t/2)+ D sinh(t/2)$, $\displaystyle y'(t)= (C/2)sinh(t/2)+ (D/2)cosh(t/2)$ so y(0)= C= 3 and y'(0)= D/2= 1/2.
    changed from "math /math" to "tex /tex" delimiters
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    Re: Particular Solution of the differential equation

    Sorry, I was unable to attach the image of my workings.. For some reason I thought the root was a complex number (j0.5). Which resulted in using the solution y= e^ αx (Acosβx+BsinβX). I now see it not a complex number to the general solution is what Romsek provided.

    Thanks!
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    Re: Particular Solution of the differential equation

    Quote Originally Posted by mrdannydandan View Post
    Sorry, I was unable to attach the image of my workings.. For some reason I thought the root was a complex number (j0.5). Which resulted in using the solution y= e^ αx (Acosβx+BsinβX). I now see it not a complex number to the general solution is what Romsek provided.

    Thanks!
    what HallsOfIvy provided. I just got his LaTex working.
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  7. #7
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    Re: Particular Solution of the differential equation

    oh yes... Thank you HallsOfIvy
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