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The given problem is an initial value problem (IVP) for a second-order differential equation. The equation is:

\[
y'' + 49y = 0
\]
with the initial conditions:
\[
y(0) = -7, \quad y'(0) = 9.
\]

### Step-by-step solution:

1. **Solve the homogeneous differential equation**:
   The characteristic equation associated with the differential equation \(y'' + 49y = 0\) is:
   \[
   r^2 + 49 = 0 \quad \Rightarrow \quad r^2 = -49 \quad \Rightarrow \quad r = \pm 7i.
   \]
   Since the roots are purely imaginary, the general solution to the differential equation is:
   \[
   y(x) = C_1 \cos(7x) + C_2 \sin(7x),
   \]
   where \(C_1\) and \(C_2\) are constants to be determined using the initial conditions.

2. **Apply the initial condition \(y(0) = -7\)**:
   Substituting \(x = 0\) into the general solution:
   \[
   y(0) = C_1 \cos(0) + C_2 \sin(0) = C_1.
   \]
   Therefore, \(C_1 = -7\).

3. **Apply the initial condition \(y'(0) = 9\)**:
   First, compute the derivative of the general solution:
   \[
   y'(x) = -7C_1 \sin(7x) + 7C_2 \cos(7x).
   \]
   Substituting \(x = 0\) into the derivative:
   \[
   y'(0) = -7C_1 \sin(0) + 7C_2 \cos(0) = 7C_2.
   \]
   Since \(y'(0) = 9\), we get:
   \[
   7C_2 = 9 \quad \Rightarrow \quad C_2 = \frac{9}{7}.
   \]

4. **Write the final solution**:
   Substituting the values of \(C_1\) and \(C_2\) into the general solution, we get the particular solution to the IVP:
   \[
   y(x) = -7 \cos(7x) + \frac{9}{7} \sin(7x).
   \]

Thus, the solution to the initial value problem is:

\[
y(x) = -7 \cos(7x) + \frac{9}{7} \sin(7x).
\]

---

Would you like more details or have any questions on the process? Here are some related problems:

1. What would the solution look like if the initial condition \( y(0) \) were different?
2. How would you solve if the equation was \( y'' + ky = 0 \) for a different constant \( k \)?
3. What happens if the characteristic equation has real roots instead of complex ones?
4. Can you verify the solution by plugging it back into the original differential equation?
5. How would the solution change if the differential equation had a non-homogeneous term?

**Tip**: When solving differential equations with complex roots, always express the solution in terms of sine and cosine for real-valued solutions.

Solve the initial value problem: y'' + 49y = 0, y(0) = -7, y'(0) = 9. Use x as the independent variable.

Learn how to solve a second-order differential equation with initial conditions: y'' + 49y = 0, y(0) = -7, y'(0) = 9. The problem involves solving using the characteristic equation, trigonometric functions, and applying initial conditions. Suitable for undergraduate calculus students.

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