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resolver la ecuacion de coeficientes constantes

The equation in the image is:

\[
\frac{d^4x}{dt^4} + 10 \frac{d^2x}{dt^2} + 9x = 0
\]

This is a **linear homogeneous differential equation with constant coefficients**. To solve it, we follow these steps:

### 1. Characteristic Equation:
We assume a solution of the form \( x(t) = e^{rt} \), where \( r \) is a constant. Substituting this into the differential equation leads to the **characteristic equation**:

\[
r^4 + 10r^2 + 9 = 0
\]

This is a **quadratic equation** in terms of \( r^2 \). Let’s solve for \( r^2 \).

### 2. Solve for \( r^2 \):

\[
r^4 + 10r^2 + 9 = 0 \quad \text{(set } y = r^2\text{)}
\]
\[
y^2 + 10y + 9 = 0
\]

Using the quadratic formula:
\[
y = \frac{-10 \pm \sqrt{10^2 - 4(1)(9)}}{2(1)} = \frac{-10 \pm \sqrt{100 - 36}}{2} = \frac{-10 \pm \sqrt{64}}{2} = \frac{-10 \pm 8}{2}
\]

Thus, the roots for \( y = r^2 \) are:
\[
y_1 = -1 \quad \text{and} \quad y_2 = -9
\]

Since \( r^2 = y \), and the values are negative, we can write the roots as:
\[
r_1 = i \quad r_2 = -i \quad r_3 = 3i \quad r_4 = -3i
\]

### 3. General Solution:
Since the roots are purely imaginary, the general solution will involve sines and cosines. For each pair of roots, we get solutions involving both sine and cosine functions. The general solution is:

\[
x(t) = C_1 \cos(t) + C_2 \sin(t) + C_3 \cos(3t) + C_4 \sin(3t)
\]

where \( C_1, C_2, C_3, C_4 \) are constants determined by initial conditions (if provided).

---

Would you like further details, or do you have any questions? Here are some related topics for exploration:

1. What are homogeneous vs. non-homogeneous differential equations?
2. How do complex roots in characteristic equations affect the solution?
3. How does the solution change if initial conditions are provided?
4. What is the significance of imaginary roots in physical systems?
5. How does this method generalize to higher-order differential equations?

**Tip:** For second-order differential equations, the general solution structure remains similar, but you only work with \( r^2 \) terms directly, simplifying the process.

Learn how to solve the fourth-order linear homogeneous differential equation d^4x/dt^4 + 10d^2x/dt^2 + 9x = 0 using characteristic equations. This solution explores imaginary roots and provides a general solution involving sines and cosines.

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