The differential equation of the family of curves ` y= c_(1)e^(x) +c_(2)e^(-x)` is

Options

A. ` (d^(2)y)/(dx^(2)) +y=0`
B. `(d^(2)y)/(dx^(2)) -y =0`
C. `(d^(2)y)/(dx^(2))+1=0`
D. `(d^(2)y)/(dx^(2)) -1 =0`

Answer

To find the differential equation of the family of curves given by \( y = c_1 e^x + c_2 e^{-x} \), we will follow these steps:

### Step 1: Differentiate the given equation
We start by differentiating the equation with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx}(c_1 e^x + c_2 e^{-x}) \]

Using the chain rule, we get:

\[ \frac{dy}{dx} = c_1 e^x - c_2 e^{-x} \]

### Step 2: Differentiate again to find the second derivative
Next, we differentiate \( \frac{dy}{dx} \) to find the second derivative.

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(c_1 e^x - c_2 e^{-x}) \]

Again applying the chain rule, we have:

\[ \frac{d^2y}{dx^2} = c_1 e^x + c_2 e^{-x} \]

### Step 3: Substitute \( y \) into the second derivative
From the original equation, we know that:

\[ y = c_1 e^x + c_2 e^{-x} \]

Thus, we can substitute \( y \) into the expression for \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = y \]

### Step 4: Rearranging to form the differential equation
Now we can rearrange the equation:

\[ \frac{d^2y}{dx^2} - y = 0 \]

This is the required differential equation of the family of curves.

### Final Answer
The differential equation of the family of curves \( y = c_1 e^x + c_2 e^{-x} \) is:

\[ \frac{d^2y}{dx^2} - y = 0 \]

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