The differential equation which represents the family of curves `y=c_(1)e^(c_(2)x), c_(1) and c_(2)` are constants is

Options

A. `y'=y^(2)`
B. `y''=y'y`
C. `y y''=y'`
D. `y y''=(y')^(2)`

Answer

To derive the differential equation representing the family of curves \( y = c_1 e^{c_2 x} \), where \( c_1 \) and \( c_2 \) are constants, we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ y = c_1 e^{c_2 x} \]
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = c_1 c_2 e^{c_2 x} \]
Since \( y = c_1 e^{c_2 x} \), we can substitute \( y \) into the equation:
\[ \frac{dy}{dx} = c_2 y \]

### Step 2: Differentiate again to find the second derivative
Now, we differentiate \( \frac{dy}{dx} = c_2 y \) again with respect to \( x \):
\[ \frac{d^2y}{dx^2} = c_2 \frac{dy}{dx} \]
Substituting \( \frac{dy}{dx} = c_2 y \) into this equation gives:
\[ \frac{d^2y}{dx^2} = c_2 (c_2 y) = c_2^2 y \]

### Step 3: Rearranging the equation
We can rearrange this equation to express it in a standard form:
\[ \frac{d^2y}{dx^2} - c_2^2 y = 0 \]

### Step 4: Expressing \( c_2 \) in terms of \( y \) and \( \frac{dy}{dx} \)
From the first derivative, we have:
\[ c_2 = \frac{\frac{dy}{dx}}{y} \]
Substituting \( c_2 \) into the second derivative equation gives:
\[ \frac{d^2y}{dx^2} - \left(\frac{\frac{dy}{dx}}{y}\right)^2 y = 0 \]
This simplifies to:
\[ \frac{d^2y}{dx^2} - \frac{(\frac{dy}{dx})^2}{y} = 0 \]

### Final Form
Thus, the differential equation that represents the family of curves \( y = c_1 e^{c_2 x} \) is:
\[ y \frac{d^2y}{dx^2} - \left(\frac{dy}{dx}\right)^2 = 0 \]