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Math Help - initial value problem O D E

  1. #1
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    initial value problem O D E

    I have read the theory but I don't understand how to use is.
    Don't know where to put the 2 before y'' when I'm solving and can't really solve it. If you have an hyperlink with more explanations and examples of solutions of ODE's I would also appreciate it.
    Thanks!

    2y''+3y'+y=x^{2}+3\sin x , y(0)=12.5 , y'(0)=-6
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  2. #2
    Super Member Showcase_22's Avatar
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    <br />
2y''+3y'+y=x^{2}+3\sin x , y(0)=12.5 , y'(0)=-6<br />
    Solve the homogeneous equation first. ie solve:

    2\lambda^2+3\lambda+1=0

    This can be done using the auxiliay equation method.

    Next, you will need to choose a complimentary function. I would try something like y=Ax^2+3sin(x)+3cos(x), but these questions have a habit of being a little tricky. You may need to modify this =S

    Finally, you can just substitute the given values back into the equation. For y'(0)=-6 you will have to differentiate your general solution.
    Normally,the two initial conditions give two simultaneous equations for A and B.

    Two simultaneous equations and two unknowns ftw!

    Anyway, post back how you get on.
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  3. #3
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    Labda can be -1, but also -1/2
    How can I use both? Or do I have to chose?
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  4. #4
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    Quote Originally Posted by Showcase_22 View Post
    Solve the homogeneous equation first. ie solve:

    2\lambda^2+3\lambda+1=0
    Quote Originally Posted by miepie View Post
    Labda can be -1, but also -1/2
    How can I use both? Or do I have to chose?
    The characteristic equation that Showcase posted comes from looking for a slution of the complementary equation

    2y'' + 3y' + y = 0,

    in the form

    y = e^{\lambda x}.

    As you mention, there are two solutions and thus, the solution of the complementary equation is

    y = c_1 e^{-\frac{1}{2}x} + c_2 e^{-x} (you want both).
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