Find it's differential equation which general solution is below:
y=(ae^(-x))+(b/x)

To find the differential equation corresponding to the general solution \( y = ae^{-x} + \frac{b}{x} \), we can follow these steps:

### Step 1: Identify the General Solution
Given:
\[
y = ae^{-x} + \frac{b}{x}
\]
where \( a \) and \( b \) are arbitrary constants.

### Step 2: Compute the First Derivative \( \frac{dy}{dx} \)
Differentiate \( y \) with respect to \( x \):
\[
\frac{dy}{dx} = \frac{d}{dx}\left( ae^{-x} + \frac{b}{x} \right)
\]
\[
\frac{dy}{dx} = a \cdot \frac{d}{dx}(e^{-x}) + b \cdot \frac{d}{dx}\left(\frac{1}{x}\right)
\]
\[
\frac{dy}{dx} = -ae^{-x} - \frac{b}{x^2}
\]

### Step 3: Compute the Second Derivative \( \frac{d^2y}{dx^2} \)
Differentiate \( \frac{dy}{dx} \) again with respect to \( x \):
\[
\frac{d^2y}{dx^2} = \frac{d}{dx}\left(-ae^{-x} - \frac{b}{x^2}\right)
\]
\[
\frac{d^2y}{dx^2} = -a \cdot \frac{d}{dx}(e^{-x}) - b \cdot \frac{d}{dx}\left(\frac{1}{x^2}\right)
\]
\[
\frac{d^2y}{dx^2} = ae^{-x} + \frac{2b}{x^3}
\]

### Step 4: Eliminate Constants \( a \) and \( b \)
To find the differential equation, express \( a \) and \( b \) in terms of \( y \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \).

From the first derivative:
\[
-ae^{-x} = \frac{dy}{dx} + \frac{b}{x^2}
\]

From the second derivative:
\[
ae^{-x} = \frac{d^2y}{dx^2} - \frac{2b}{x^3}
\]

Adding the two equations:
\[
ae^{-x} - ae^{-x} = \frac{d^2y}{dx^2} - \frac{2b}{x^3} + \frac{dy}{dx} + \frac{b}{x^2}
\]
\[
0 = \frac{d^2y}{dx^2} + \frac{dy}{dx} + b\left(-\frac{2}{x^3} + \frac{1}{x^2}\right)
\]

Since \( b \) is arbitrary, the terms with \( b \) must cancel out, leading to:
\[
\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0
\]

Thus, the differential equation is:
\[
\boxed{\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0}
\]

Would you like more details on this process or have any questions?

Here are five related questions:
1. How do you find the differential equation for a given specific solution?
2. What is the role of arbitrary constants in forming differential equations?
3. Can you solve the differential equation \(\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\) using characteristic equations?
4. How does one verify that a given function is a solution to a differential equation?
5. What are the common techniques for solving second-order linear homogeneous differential equations?

**Tip:** Understanding how to derive differential equations from general solutions is crucial for solving boundary value problems in physics and engineering.

Discover how to derive the differential equation corresponding to the general solution y = ae^(-x) + b/x step-by-step. Learn differentiation and integration techniques relevant to solving differential equations. Suitable for undergraduate students and those studying differential equations.

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