The growth of a quantity N(t) at any instant t is given by ` (dN(t))/(dt) = alphaN(t)` , Given that `N(t)=ce^(kt)` , is constant. What is the value of `alpha` ?

Options

A. c
B. k
C. c+k
D. c-k

Explanation

To find the value of \( \alpha \) in the differential equation given by

\[ \frac{dN(t)}{dt} = \alpha N(t) \]

where \( N(t) = ce^{kt} \) (with \( c \) being a constant), we will follow these steps:

### Step 1: Differentiate \( N(t) \)

First, we need to differentiate \( N(t) = ce^{kt} \) with respect to \( t \).

\[ \frac{dN(t)}{dt} = \frac{d}{dt}(ce^{kt}) \]

Using the chain rule, we get:

\[ \frac{dN(t)}{dt} = c \cdot k e^{kt} \]

### Step 2: Substitute \( N(t) \) into the differential equation

Now, we substitute \( N(t) \) into the original differential equation:

\[ \frac{dN(t)}{dt} = \alpha N(t) \]

Substituting \( N(t) \):

\[ c \cdot k e^{kt} = \alpha (ce^{kt}) \]

### Step 3: Simplify the equation

We can simplify the equation by dividing both sides by \( ce^{kt} \) (assuming \( c \neq 0 \) and \( e^{kt} \neq 0 \)):

\[ k = \alpha \]

### Conclusion

Thus, we find that:

\[ \alpha = k \]