The differential equaiotn which represents the family of curves `y=C_(1)e^(C_(2)x)`, where `C_(1)` and `C_(2)` are arbitrary constants, is

Options

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

Explanation

To find the differential equation that represents the family of curves given by \( y = C_1 e^{C_2 x} \), where \( C_1 \) and \( C_2 \) are arbitrary constants, we will follow these steps:

### Step 1: Differentiate the given equation with respect to \( x \)

Given:
\[ y = C_1 e^{C_2 x} \]

Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = C_1 C_2 e^{C_2 x} \]

### Step 2: Express \( C_1 \) in terms of \( y \) and \( e^{C_2 x} \)

From the original equation, we can express \( C_1 \) as:
\[ C_1 = \frac{y}{e^{C_2 x}} \]

### Step 3: Substitute \( C_1 \) into the derivative

Substituting \( C_1 \) into the derivative:
\[ \frac{dy}{dx} = C_2 \left( \frac{y}{e^{C_2 x}} \right) e^{C_2 x} \]
This simplifies to:
\[ \frac{dy}{dx} = C_2 y \]

### Step 4: Differentiate again to find the second derivative

Now, differentiate \( \frac{dy}{dx} = C_2 y \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = C_2 \frac{dy}{dx} \]

### Step 5: Substitute \( \frac{dy}{dx} \) back into the second derivative

Substituting \( \frac{dy}{dx} = C_2 y \) into the second derivative:
\[ \frac{d^2y}{dx^2} = C_2 (C_2 y) = C_2^2 y \]

### Step 6: Formulate the differential equation

Now we have:
\[ \frac{d^2y}{dx^2} = C_2^2 y \]

To eliminate \( C_2 \), we can express \( C_2 \) in terms of \( \frac{dy}{dx} \):
From \( \frac{dy}{dx} = C_2 y \), we can write:
\[ C_2 = \frac{\frac{dy}{dx}}{y} \]

Substituting this into the second derivative equation:
\[ \frac{d^2y}{dx^2} = \left( \frac{\frac{dy}{dx}}{y} \right)^2 y \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = \frac{1}{y} \left( \frac{dy}{dx} \right)^2 \]

### Final Differential Equation

Thus, the final differential equation representing the family of curves is:
\[ y \frac{d^2y}{dx^2} - \left( \frac{dy}{dx} \right)^2 = 0 \]

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