The differential equation of `y= 2be^(ax)` (a & b are parameters) is

Options

A. `y y_(1)= y_(2)^(2)`
B. `y y_(2)= y_(1)^(2)`
C. `y y_(1)^(2) = y_(2)`
D. `y y_(2)^(2)= y_(1)`

Answer

To find the differential equation of the function \( y = 2b e^{ax} \), we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)

Given:
\[ y = 2b e^{ax} \]

We differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2b \cdot \frac{d}{dx}(e^{ax}) = 2b \cdot a e^{ax} = 2ab e^{ax} \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2ab e^{ax}) = 2ab \cdot \frac{d}{dx}(e^{ax}) = 2ab \cdot a e^{ax} = 2a^2b e^{ax} \]

### Step 3: Formulate the differential equation

We have:
- \( y = 2b e^{ax} \)
- \( \frac{dy}{dx} = 2ab e^{ax} \)
- \( \frac{d^2y}{dx^2} = 2a^2b e^{ax} \)

From the second derivative, we can express \( b \) in terms of \( y \):
\[ b = \frac{y}{2e^{ax}} \]

Substituting \( b \) in \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = 2a^2 \left(\frac{y}{2e^{ax}}\right) e^{ax} = a^2 y \]

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

### Final Result

The differential equation of \( y = 2b e^{ax} \) is:
\[ \frac{d^2y}{dx^2} - a^2 y = 0 \]

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