https://file.aiquickdraw.com/mathbot/file/1726070504421vtcd1xap.jpg

To solve the given problem, we need to check which of the provided functions satisfies the second-order differential equation:

\[
\frac{\partial^2 y}{\partial x^2} - 9 \frac{\partial y}{\partial x} = 0.
\]

### Let's break this down:

1. **Step 1**: The differential equation involves the second derivative \(\frac{\partial^2 y}{\partial x^2}\) and the first derivative \(\frac{\partial y}{\partial x}\). We need to find these derivatives for each of the given options.

2. **Step 2**: Substitute the function into the equation and check which one satisfies the differential equation.

### Let's evaluate the derivatives for each option:

- **Option A**: \(y = e^{-9x}\)
  - First derivative: \(\frac{dy}{dx} = -9e^{-9x}\)
  - Second derivative: \(\frac{d^2y}{dx^2} = 81e^{-9x}\)
  - Substitute into the differential equation:
    \[
    81e^{-9x} - 9(-9e^{-9x}) = 81e^{-9x} + 81e^{-9x} = 0.
    \]
    **This does not hold.**

- **Option B**: \(y = e^{-3x}\)
  - First derivative: \(\frac{dy}{dx} = -3e^{-3x}\)
  - Second derivative: \(\frac{d^2y}{dx^2} = 9e^{-3x}\)
  - Substitute into the differential equation:
    \[
    9e^{-3x} - 9(-3e^{-3x}) = 9e^{-3x} + 27e^{-3x} = 0.
    \]
    **This does not hold.**

- **Option C**: \(y = e^{9x}\)
  - First derivative: \(\frac{dy}{dx} = 9e^{9x}\)
  - Second derivative: \(\frac{d^2y}{dx^2} = 81e^{9x}\)
  - Substitute into the differential equation:
    \[
    81e^{9x} - 9(9e^{9x}) = 81e^{9x} - 81e^{9x} = 0.
    \]
    **This holds true.**

- **Option D**: \(y = e^{3x}\)
  - First derivative: \(\frac{dy}{dx} = 3e^{3x}\)
  - Second derivative: \(\frac{d^2y}{dx^2} = 9e^{3x}\)
  - Substitute into the differential equation:
    \[
    9e^{3x} - 9(3e^{3x}) = 9e^{3x} - 27e^{3x} = 0.
    \]
    **This does not hold.**

### Conclusion:
The correct answer is **C: \(y = e^{9x}\)**, as it satisfies the differential equation.

---

Would you like more details on any of the steps? Here are five related questions:

1. How do you find the first and second derivatives of exponential functions?
2. What is a general solution to second-order linear homogeneous differential equations?
3. Why do exponential functions often appear in differential equations?
4. How can you solve a second-order differential equation with constant coefficients?
5. What are the applications of solving such differential equations?

**Tip:** In solving differential equations, verifying each step carefully with correct derivatives is essential to ensure the solution satisfies the equation.

Which of the following functions will satisfy the differential equation (d^2y/dx^2) - 9(dy/dx) = 0?

Learn how to solve the second-order differential equation (d^2y/dx^2) - 9(dy/dx) = 0 by checking which exponential function satisfies the equation. Detailed steps for solving this type of equation are included, suitable for high school and early university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation