91Ó°ÊÓ

In the context of differential equations, the auxiliary equation (or characteristic equation) is essential for solving higher-order linear homogeneous differential equations. This equation is derived directly from the differential equation by assuming a trial solution of the form \(y = e^{rx}\), where \(r\) represents the roots we need to find.
For a differential equation with constant coefficients, say of the form \(a_n y^{(n)} + a_{n-1} y^{(n-1)} + \[...\] + a_1 y' + a_0 y = 0\), we substitute \(y = e^{rx}\) and its derivatives back into this equation. This substitution transforms the problem into finding the roots of the polynomial \(a_n r^n + a_{n-1} r^{n-1} + \[...\] + a_1 r + a_0 = 0\).

For the given exercise, the sixth-order equation gives us six roots: 7, 9, and the pairs \(2 \pm 4 i\). Solving the auxiliary equation helps us in forming the general solution to the differential equation.

Complex conjugate roots appear in pairs, such as \(a \pm bi\). When these roots are solutions to the auxiliary equation, they contribute terms that combine exponential and trigonometric functions to the general solution.
For the complex conjugate pair \(2 \pm 4i\) in our problem, we use terms like \(e^{2x}\cos(4x)\) and \(e^{2x}\sin(4x)\).

In cases where you have repeated complex conjugate roots, each repetition adds multiplicative factors of \(x\). For example, since \(2 \pm 4i\) appears twice, the general form will include terms multiplied by \(x\), yielding: \(C_3 e^{2x} \cos(4x)\), \(C_4 e^{2x} \sin(4x)\), \(C_5 x e^{2x} \cos(4x)\), and \(C_6 x e^{2x} \sin(4x)\). This ensures we account for all occurrences of these roots.

The general solution to a linear homogeneous differential equation summarizes all possible solutions derived from the roots of the auxiliary equation.
When dealing with real roots, each root \(r_i\) contributes a term \(C_i e^{r_i x}\) to the solution. For our problem, the real roots 7 and 9 contribute \(C_1 e^{7x}\) and \(C_2 e^{9x}\).

Complex conjugate roots, such as \(2 \pm 4i\), add terms combining exponentials and trigonometric functions, specifically \(e^{ax} \cos(bx)\) and \(e^{ax} \sin(bx)\). For repeated complex roots, factors of \(x\) are included to account for multiplicity. Hence, for \(2 \pm 4i\) repeated twice, we get \(C_3 e^{2x} \cos(4x)\), \(C_4 e^{2x} \sin(4x)\), \(C_5 x e^{2x} \cos(4x)\), and \(C_6 x e^{2x} \sin(4x)\).

Putting it all together, the comprehensive general solution to the provided sixth-order differential equation is:
\[y(x) = C_1 e^{7x} + C_2 e^{9x} + C_3 e^{2x} \cos(4x) + C_4 e^{2x} \sin(4x) + C_5 x e^{2x} \cos(4x) + C_6 x e^{2x} \sin(4x)\].