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Math Help - Second order differential equations

  1. #1
    Member roshanhero's Avatar
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    Second order differential equations

    What should I try for the particular integral inorder to solve these problems where D means d/dx.
    1.(D^2-4D+4)y=x^3e^{2x}
    2.\frac{d^2y}{dx^2}+4y=sin^2x
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    Senior Member Peritus's Avatar
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  3. #3
    Member roshanhero's Avatar
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    But I couldnot get it still.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by roshanhero View Post
    What should I try for the particular integral inorder to solve these problems where D means d/dx.
    1.(D^2-4D+4)y=x^3e^{2x}
    Are you familiar with the so-called "Annihilator" approach?? To see how you do it this way, look at post #7 in my differential equations tutorial [also look at post #6 and #8 for two other techniques].

    Going along with this, we solve the homogeneous solution first:

    You should end up with y_c=c_1e^{2x}+c_2xe^{2x}

    Now for the particular solution, solve the non homogeneous equation.

    This will lead us to finding the annihilator of x^3e^{2x}, which is \left(D-2\right)^4

    The DE becomes \left(D-2\right)^2\left(D-2\right)^4=0\implies (r-2)^6=0\implies r=2 with multiplicity six. Thus, the particular solution with only consist of r repeating 4 times, since 2 were used in the homogeneous solution.

    Thus, y_p=Ax^2e^{2x}+Bx^3e^{2x}+Cx^4e^{2x}+Dx^5e^{2x}

    Now substitute y_p into the original DE and find the coefficients...

    2.\frac{d^2y}{dx^2}+4y=sin^2x
    If you still do the annihilator approach, \frac{d^2y}{dx^2}+4y=\sin^2x\implies \left(D^2+4\right)y=\tfrac{1}{2}-\tfrac{1}{2}\cos (2x)

    Try to do something similar to what I have done above.

    I hope this helps.

    --Chris
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