Question

Solve the following differential equation?

(D-1)^(2)[D^(2)+1)^(2)y=sin^(2)(x/2)+e^(x)+x

Answer 1

`y = (Ax+B)sin(x) + (Cx+D)cos(x) + (Ex+F)e^x + 1/8x^2e^x - 1/32 x^2sin(x) + x + 5/2`

for constants `A, B, C, D, E, and F`

Explanation:

We can rewrite this as
`((D-1)(D^2+1))^2y = 1/2 - 1/2 cos(x) + e^x + x`

Just looking at the left side,
`(D^3 - D^2 + D - 1)^2y`
`= (D^6 - 2D^5 + 3D^4-4D^3+3D^2-2D + 1)y = Oy`
(where we define `O` to be our operator for ease later)

Let's focus on the specific solution before we discuss homogenous solutions. It is logical to assume some functional forms of our solutions.

For the constant, clearly that constant can be our function, i.e.
`O(a) = a \ \ forall a in mathbb(C)`.

For a linear function, all higher order derivatives will be negligible, so we can try a form of `y_0 = Ax + B`. Plugging this in,
`Oy_0 = -2A + Ax+B = Ax + (B-2A)`

To get the constants above, therefore `A = 1` and `B = 5/2`.

For an exponential function `y_1 = Ce^x`, `D` is an identity, hence
`Oy_1 = (C - 2C + 3C-4C+3C-2C+C)e^x = 0`

Therefore, we try `y_2 = Fxe^x`, which is a little less lovely until you realize that the `n`th derivative of `xe^x` equals `xe^x + n e^x` (try it!).

Therefore (the `xe^x` terms are identical to before and will therefore cancel)
`Oy_2 = 6e^x - 10e^x + 12e^x - 12e^x + 6e^x - 2e^x = 0`

Therefore, we will try `y_3 = Gx^2e^x`. Here, one can prove that
`d^n/dx^n(x^2e^x) = x^2e^x +2nxe^x + n(n-1)e^x`

Therefore, the first two terms will cancel out again, and
`Oy_3 = G(30 - 2 * 20 + 3 * 12 - 4 * 6 + 3 * 2) = 8G`
Therefore, `G = 1/8`. This need for `x^2` is obvious at the factored version of the differential equation.

We can repeat this process for the sinusoid and we find that we need to use `-1/32 x^2sin(x)`. Therefore, the specific solution is
`y_c = 1/8x^2e^x - 1/32 x^2sin(x) + x + 5/2`

As is apparent by the factored differential equation, we have eigenfunctions of `sin(x)`, `cos(x)`, `e^x`, and each of those multiplied by `x`.

Therefore, our general solution is
`y = (Ax+B)sin(x) + (Cx+D)cos(x) + (Ex+F)e^x + y_c`