To find a particular solution $y_p(x)$ for the differential equation

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

we need to choose a form for $y_p(x)$ based on the form of the non-homogeneous term, $8x^2 - 3$.

Step-by-Step Solution

1. Identify the Non-Homogeneous Term: The right side of the equation, $8x^2 - 3$, is a polynomial of degree 2.

2. Form of the Particular Solution: Since the non-homogeneous term is a polynomial, we assume a particular solution of the same form. Let: $y_p(x) = Ax^2 + Bx + C,$ where $A$, $B$, and $C$ are constants to be determined.

3. Compute Derivatives of $y_p(x)$:

   • $y_p''(x) = 2A$,
   • $y_p^{(3)} = 0$,
   • $y_p^{(5)} = 0$.

4. Substitute into the Differential Equation: Substitute $y_p(x)$, $y_p^{(3)}(x)$, and $y_p''(x)$ into the differential equation: $y^{(5)} + 2y^{(3)} + 2y'' = 0 + 2 \cdot 0 + 2 \cdot (2A) = 4A.$

5. Set up the Equation: The equation becomes: $4A = 8x^2 - 3.$

Since the expression on the left side must match the entire right side, we need to solve for the values of $A$, $B$, and $C$ that satisfy this relationship. However, since we only have constants on the left side, this suggests re-evaluating the assumption or directly setting values.

Would you like a breakdown of each calculation step or further detail in this solution process?

Relative Questions:

1. How does the degree of the polynomial on the right side affect the choice of the particular solution?
2. What would change if the non-homogeneous term were an exponential or trigonometric function?
3. How do the characteristics of the homogeneous solution impact the choice of the particular solution?
4. What is the purpose of finding a particular solution in solving differential equations?
5. Can this method be used for higher-order differential equations with more complex right-hand terms?

Tip: Always match the form of the particular solution to the type of function in the non-homogeneous term, considering any duplication with terms in the complementary (homogeneous) solution.