The equation you've provided is a second-order linear homogeneous differential equation:

\[
x'' + 3x - 10 = 0
\]

To solve this, we'll find the general solution by solving the characteristic equation.

### Step 1: Write the characteristic equation
For a differential equation of the form:

\[
x'' + ax' + bx = 0
\]

The characteristic equation is:

\[
r^2 + ar + b = 0
\]

In this case, since there's no \(x'\) term, \(a = 0\) and \(b = 3\):

\[
r^2 + 3r - 10 = 0
\]

### Step 2: Solve the characteristic equation
To find the roots \(r\), we use the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values \(a = 1\), \(b = 3\), and \(c = -10\):

\[
r = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)}
\]
\[
r = \frac{-3 \pm \sqrt{9 + 40}}{2}
\]
\[
r = \frac{-3 \pm \sqrt{49}}{2}
\]
\[
r = \frac{-3 \pm 7}{2}
\]

So, the roots are:

\[
r_1 = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad r_2 = \frac{-3 - 7}{2} = -5
\]

### Step 3: Write the general solution
Since the characteristic equation has two distinct real roots, the general solution to the differential equation is:

\[
x(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}
\]

Substituting the values of \(r_1\) and \(r_2\):

\[
x(t) = C_1 e^{2t} + C_2 e^{-5t}
\]

### Final Answer:

The general solution to the differential equation \(x'' + 3x - 10 = 0\) is:

\[
x(t) = C_1 e^{2t} + C_2 e^{-5t}
\]

Would you like more details on any of these steps, or do you have any other questions?

---

Here are some related questions you might find interesting:
1. How do you solve non-homogeneous differential equations?
2. What is the physical meaning of the roots of the characteristic equation?
3. How do you determine the constants \(C_1\) and \(C_2\) in a general solution?
4. What happens if the characteristic equation has complex roots?
5. How do you solve differential equations with higher-order derivatives?

**Tip:** The nature of the roots of the characteristic equation (real, repeated, or complex) determines the form of the general solution for the differential equation.

Learn how to solve the second-order linear homogeneous differential equation x'' + 3x - 10 = 0 with a detailed step-by-step solution. This problem covers differential equations, quadratic formulas, and theorems related to existence and uniqueness. Suitable for advanced high school students.

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