To solve the second-order differential equation:

\[
y'' + 10y' + 74y = 0
\]

we use the characteristic equation method.

### Step 1: Write the characteristic equation
The given differential equation can be written in terms of a characteristic equation by assuming a solution of the form \( y = e^{rt} \), where \( r \) is a constant. Substituting \( y = e^{rt} \) into the differential equation:

\[
r^2 e^{rt} + 10r e^{rt} + 74 e^{rt} = 0
\]

Dividing through by \( e^{rt} \) (which is never zero) yields the characteristic equation:

\[
r^2 + 10r + 74 = 0
\]

### Step 2: Solve the characteristic equation
This is a quadratic equation. We can solve it using the quadratic formula:

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

For our equation \( r^2 + 10r + 74 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 10 \)
- \( c = 74 \)

Now, substitute these values into the quadratic formula:

\[
r = \frac{-10 \pm \sqrt{10^2 - 4(1)(74)}}{2(1)}
\]
\[
r = \frac{-10 \pm \sqrt{100 - 296}}{2}
\]
\[
r = \frac{-10 \pm \sqrt{-196}}{2}
\]
\[
r = \frac{-10 \pm 14i}{2}
\]
\[
r = -5 \pm 7i
\]

### Step 3: Write the general solution
Since the roots are complex, \( r = -5 \pm 7i \), the general solution to the differential equation is:

\[
y(t) = e^{-5t} \left( C_1 \cos(7t) + C_2 \sin(7t) \right)
\]

where \( C_1 \) and \( C_2 \) are arbitrary constants determined by initial conditions.

Let me know if you'd like further details or have questions! Here are some related questions for deeper understanding:

1. How would you solve if the characteristic equation had real roots?
2. What is the significance of the imaginary part of the solution?
3. How do initial conditions affect the solution of the differential equation?
4. How would you solve a non-homogeneous differential equation with similar terms?
5. What physical systems can be modeled by this type of differential equation?

**Tip:** Complex roots in characteristic equations lead to oscillatory solutions involving sine and cosine functions.

Learn how to solve the second-order differential equation y'' + 10y' + 74 = 0 using the characteristic equation method. This solution includes solving for complex roots and constructing the general solution with sine and cosine terms. Ideal for advanced high school and early college students.

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