To find a particular solution ($y_p$) for the given equation:

$y^{(5)} + 2y^{(3)} + 2y'' = 8x^2 - 3$

we will use the method of undetermined coefficients.

Step 1: Analyze the right-hand side

The non-homogeneous term on the right-hand side is $8x^2 - 3$, which is a polynomial of degree 2. For polynomials, the assumed form of $y_p$ is another polynomial of the same degree as the right-hand side. Hence, let:

$y_p = Ax^2 + Bx + C$

where $A$, $B$, and $C$ are constants to be determined.

Step 2: Compute derivatives of $y_p$

$y_p' = 2Ax + B, \quad y_p'' = 2A, \quad y_p^{(3)} = 0, \quad y_p^{(5)} = 0$

Substitute these derivatives into the original equation.

Step 3: Substitute into the differential equation

Substituting into:

$y^{(5)} + 2y^{(3)} + 2y'' = 8x^2 - 3,$

we get:

$0 + 2(0) + 2(2A) = 8x^2 - 3.$

Simplify:

$4A = 8x^2 - 3.$

Clearly, this equation only matches the constant term ($-3$) and the $x^2$-coefficient, which implies $A$ is determined solely by (8x^2 such easily matches observations for manually-testing simpler-layer scenarios for degree o