	The differential equation satisfied by family of curves y=Ae^x+Be^3x+Ce^5x  where A,B,C are "arbitrary constants is:" 
A. (d^3 y)/(dx^3 )-9 (d^2 y)/(dx^2 )+23 dy/dx+15y=0		B. (d^3 y)/(dx^3 )+9 (d^2 y)/(dx^2 )-23 dy/dx-15y=0
C. (d^3 y)/(dx^3 )+9 (d^2 y)/(dx^2 )-23 dy/dx+15y=0		D. (d^3 y)/(dx^3 )-9 (d^2 y)/(dx^2 )+23 dy/dx-15y=0


Given the family of curves \( y = Ae^x + Be^{3x} + Ce^{5x} \), where \( A \), \( B \), and \( C \) are arbitrary constants, we need to find the differential equation that this family satisfies.

### Step 1: Compute the derivatives
Let's compute the first, second, and third derivatives of \( y \) with respect to \( x \).

1. **First derivative**:
   \[
   \frac{dy}{dx} = A e^x + 3B e^{3x} + 5C e^{5x}
   \]

2. **Second derivative**:
   \[
   \frac{d^2 y}{dx^2} = A e^x + 9B e^{3x} + 25C e^{5x}
   \]

3. **Third derivative**:
   \[
   \frac{d^3 y}{dx^3} = A e^x + 27B e^{3x} + 125C e^{5x}
   \]

### Step 2: Form a linear combination of the derivatives
To eliminate the arbitrary constants \( A \), \( B \), and \( C \), we form a linear combination of the derivatives:

Consider:
\[
\frac{d^3 y}{dx^3} - 9\frac{d^2 y}{dx^2} + 23\frac{dy}{dx} - 15y
\]

Substituting the derivatives:

\[
\left( A e^x + 27B e^{3x} + 125C e^{5x} \right) - 9\left( A e^x + 9B e^{3x} + 25C e^{5x} \right) + 23\left( A e^x + 3B e^{3x} + 5C e^{5x} \right) - 15\left( A e^x + B e^{3x} + C e^{5x} \right)
\]

Simplifying:

\[
= A \left( 1 - 9 + 23 - 15 \right) e^x + B \left( 27 - 81 + 69 - 15 \right) e^{3x} + C \left( 125 - 225 + 115 - 15 \right) e^{5x}
\]

\[
= A \cdot 0 \cdot e^x + B \cdot 0 \cdot e^{3x} + C \cdot 0 \cdot e^{5x} = 0
\]

Thus, the correct differential equation is:
\[
\frac{d^3 y}{dx^3} - 9\frac{d^2 y}{dx^2} + 23\frac{dy}{dx} - 15y = 0
\]

### Answer
The correct option is **D**.

Would you like more details or have any further questions?

Here are 8 related questions:
1. How do you find the differential equation of a family of curves?
2. Why is the order of the differential equation equal to the number of arbitrary constants in the family of curves?
3. What role do linear combinations play in eliminating arbitrary constants?
4. How do you check if a function is a solution to a differential equation?
5. Can you find the differential equation for a given set of solutions?
6. What are higher-order differential equations, and how do they differ from first-order equations?
7. How do you determine the coefficients in a differential equation derived from a given family of curves?
8. What are some examples of families of curves that lead to differential equations of higher order?

**Tip:** When finding the differential equation of a family of curves, always verify that the final equation has the correct order, which should correspond to the number of arbitrary constants in the original expression.

Explore the differential equation satisfied by the family of curves y = Ae^x + Be^3x + Ce^5x, derived step-by-step. Learn how to find the correct differential equation using derivatives and linear combinations. Suitable for advanced high school students and those interested in differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation