Solve the given differential equation $(D^2+1)^{2}y= 24 x \cos x$

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Solution 1

Since the question shows no try, so this is just a hint. Take $$y=(A+Bx)\sin(x)+(C+Dx)\cos(x)$$ and find the unknown coefficients.

Solution 2

It's a linear differential equation with constant coefficients. This way you immediately know that the homogeneous solutions will be exponential/trigonometric and if there are any duplicate roots of the characteristic equations (there are, as you can see the left side is already factorized), you will get polynomial*exponential/trigonometric, too. The particular solution also follows the familiar procedure with trial functions in this case.

However, this question looks like it's made for Laplace transform. Just substitute $D$ with $s$ on the left, find the laplace transform of the right hand side from the tables, and invert the transform (you will need the initial conditions in this step).

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Bharath Teja
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Updated on July 08, 2021

Comments

  • Bharath Teja
    Bharath Teja over 2 years

    $$(D^2+1)^{2}y= 24 x \cos x$$ given that $y=Dy=D^2y=0$ and $D^3 y=12$ when $x=0$

    A part of the solution I got but finding difficulty in the other part. Let it be y= y1 +y2 $y1=(A+Bx)sin(x)+(C+Dx)cos(x)$ For Integral part i was struck at $$y2=(1/(D^2+1)^{2})24x cosx $$ $$y2=-12i (1/(D^2+1))[(1/D-i)-(1/D+i)]xe^{ix}$$

    The answer is $y=3x^2sinx-x^3cosx$

    • orion
      orion about 5 years
      What did you try and how far did you get? We need to know where the problem is.
    • Bharath Teja
      Bharath Teja about 5 years
      This problem was asked in UPSC CSE mathematics optional paper. Its an exam for Civil services in India. I am finding difficulty in particular integral.
  • Bharath Teja
    Bharath Teja about 5 years
    This is part of solution. We need to find Particular integral too.,
  • orion
    orion about 5 years
    You see that $x \cos x$ and $x\sin x$ are already in the homogeneous solution. The procedure is then to increase the polynomial order until you get a match. Because $\sin$ and $\cos$ are not independent in this sense, you always need to include both. Try $(ax^2+bx+c)\cos x+(dx^2+fx+g)\sin x$.
  • Mikasa
    Mikasa about 5 years
    Thanks for your nice comment. +
  • Bharath Teja
    Bharath Teja about 5 years
    I edited the question please see it.