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Math Help - differential equations

  1. #1
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    differential equations

    show that y=e^-t is a solution of d^2y/dt^2-dy/dt - 2y=0.

    find the values of m for which y=e^mt is a solution of d^2y/dt^2+dy/dt -6y=0.

    find the solution of the first order initial- value problem dy/dx=2x. y(1)=2.

    Solve the initia-value problem d^2y/dx^2+ dy/dx = e^x y(0) = 1, y (0)=0.
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  2. #2
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    Quote Originally Posted by lay137 View Post
    show that y=e^-t is a solution of d^2y/dt^2-dy/dt - 2y=0.
    Simplify substitute or you can write the charachteristic equation:
    k^2-k-2=0
    Thus, k=2,-1
    Thus, the set
    C_1*e^{2t}+C_2*e^{-t} is the basis for the solutions.

    find the values of m for which y=e^mt is a solution of d^2y/dt^2+dy/dt -6y=0.
    Again, charachterstic equation,
    k^2+k-6=0
    Thus,
    k=-3,2
    Thus,
    e^{-3t} and e^{2t} are solutions,
    Thus possible values of "m" are,
    -3,2
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  3. #3
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    Quote Originally Posted by lay137 View Post
    find the solution of the first order initial- value problem dy/dx=2x. y(1)=2.
    You have,
    y'=2x
    Simpligy integrate, all solutions form the set,
    y=x^2+C
    When x=2 then y=1 (initial value),
    1=4+C thus, C=-3
    Thus,
    y=x^2-3
    Is the unique solution to this initial problem.
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  4. #4
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    Solve the initia-value problem d^2y/dx^2+ dy/dx = e^x y'(0) = 1, y (0)=0.
    The homogenous equation is,
    y''+y'=e^x
    Reduction of order let u=y' then u'=y''
    Thus,
    u'+u=e^x
    This is a first order linear differencial equation.
    Thus,
    u=1/m(x) * INTEGRAL m(x)e^x dx
    Where m(x) is integrating factor which is,
    m(x)=exp (INTEGRAL 1 dx)=e^x
    Thus, we have,
    u=e^{-x}*INTEGRAL e^{2x}dx
    Thus,
    u=e^{-x}*[(1/2)e^{2x}+C]
    Thus,
    u=(1/2)e^x+C*e^{-x}
    Then y is the integral of that,
    y=(1/2)e^x-C*e^{-x}+K

    We have from all of this that,
    y'=(1/2)e^x+C*e^{-x}
    y=(1/2)e^x-C*e^{-x}+K
    Now we substitute initial conditions,
    1=(1/2)+C thus, C=1/2
    0=(1/2)-C+K thus, K=0
    Therefore the unique solution is,
    y=(1/2)e^x+(1/2)*e^{-x}
    Last edited by ThePerfectHacker; October 19th 2006 at 07:40 AM.
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