1. **Stating the problem:** We are given the boundary value problem $$-y^{(4)} + p y = x^4 (32^2 x (-6(7 - 55 x^4 + 70 x^8) + 2 (x^2 - 3 x^6 + 2 x^{10})) \cos x + x^4 (x^4 - 1)^2 - 5 x^4 (x^4 - 1)^2 x \in [0,1] + 48^3 x^2 (7 - 33 x^4 + 30 x^8) - 240 (7 - 99 x^4 + 182 x^8)) \sin x),$$ with boundary conditions $$y(1) = 0, y'(1) = 0, y(-1) = 0, y'( -1) = 0.$$ 2. **Understanding the problem:** This is a fourth-order linear differential equation with parameter $p$ and complicated forcing terms involving polynomials and trigonometric functions. The boundary conditions specify values of $y$ and its first derivative at $x=1$ and $x=-1$. 3. **General approach:** To solve such a boundary value problem, we typically: - Find the complementary solution $y_c$ by solving the homogeneous equation $$-y^{(4)} + p y = 0.$$ - Find a particular solution $y_p$ to the nonhomogeneous equation using methods like undetermined coefficients or variation of parameters. - Apply boundary conditions to determine constants in $y_c$. 4. **Homogeneous equation:** Rewrite as $$y^{(4)} = p y.$$ The characteristic equation is $$r^4 = p.$$ The roots depend on $p$: - If $p > 0$, roots are $$r = \pm p^{1/4}, \pm i p^{1/4}.$$ - If $p < 0$, roots are complex with real and imaginary parts. 5. **Complementary solution:** For $p > 0$, $$y_c = C_1 e^{p^{1/4} x} + C_2 e^{-p^{1/4} x} + C_3 \cos(p^{1/4} x) + C_4 \sin(p^{1/4} x).$$ 6. **Particular solution:** Due to the complexity of the forcing term, involving polynomials times sine and cosine, we use the method of undetermined coefficients or variation of parameters. This involves guessing a solution form with polynomial-trigonometric terms matching the forcing function and solving for coefficients. 7. **Applying boundary conditions:** Substitute $x=1$ and $x=-1$ into $$y = y_c + y_p$$ and its derivative $$y' = y_c' + y_p'$$ to form a system of four equations for $C_1, C_2, C_3, C_4$. Solve this system to find constants. 8. **Summary:** The solution is $$y = y_c + y_p$$ with $y_c$ as above and $y_p$ found by matching the forcing term. Constants are determined by boundary conditions. This problem requires symbolic or numerical computation software for explicit solutions due to complexity. **Final answer:** The general solution form is $$y = C_1 e^{p^{1/4} x} + C_2 e^{-p^{1/4} x} + C_3 \cos(p^{1/4} x) + C_4 \sin(p^{1/4} x) + y_p(x),$$ where $y_p(x)$ is a particular solution matching the forcing term, and constants $C_1, C_2, C_3, C_4$ satisfy the boundary conditions.