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

Options

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

Answer

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 equation once
Start with the given equation:
\[ y = c_1 e^{c_2 x} \]
Differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = c_1 \cdot \frac{d}{dx}(e^{c_2 x}) = c_1 \cdot e^{c_2 x} \cdot c_2 \]
Thus, we can write:
\[ \frac{dy}{dx} = c_2 c_1 e^{c_2 x} \]

### Step 2: Substitute \( y \) into the equation
From the original equation, we know that \( y = c_1 e^{c_2 x} \). Therefore, we can express \( c_1 \) in terms of \( y \):
\[ c_1 = \frac{y}{e^{c_2 x}} \]
Substituting this into the derivative equation gives:
\[ \frac{dy}{dx} = c_2 \left(\frac{y}{e^{c_2 x}}\right) e^{c_2 x} = c_2 y \]

### Step 3: Differentiate again
Now, differentiate \( \frac{dy}{dx} = c_2 y \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = c_2 \frac{dy}{dx} \]

### Step 4: Solve for \( c_2 \)
From the first derivative, we have:
\[ c_2 = \frac{\frac{d^2y}{dx^2}}{\frac{dy}{dx}} \]

### Step 5: Substitute \( c_2 \) back into the equation
Substituting \( c_2 \) back into the first derivative equation:
\[ \frac{dy}{dx} = \left(\frac{\frac{d^2y}{dx^2}}{\frac{dy}{dx}}\right) y \]
Cross-multiplying gives:
\[ \left(\frac{dy}{dx}\right)^2 = y \frac{d^2y}{dx^2} \]

### Step 6: Final form of the differential equation
This can be rewritten as:
\[ y \frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2 \]

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