A radioactive with decay constant `lambda` is being produced in a nuclear reactor at a rate `q_(0)t` per second, where `q_(0)` is a positive constant and t is the time. During each decay, `E_(0)` energy is released. The production of radionuclide starts at time `t=0`.
Which differential equation correctly represents the above process?.

Options

A. `(dN)/(dt)+lambda N=q_(0) t`
B. `(dN)/(dt)-lambda N=q_(0) t`
C. `(dN)/(dt)+q_(0) t=lambda N`
D. `(dN)/(dt)+q_(0) t=-lambda N`

Explanation

To derive the differential equation representing the process described in the question, we will follow these steps:

### Step 1: Understand the production and decay of the radionuclide
We know that the radionuclide is being produced at a rate of \( q_0 t \) per second, where \( q_0 \) is a constant and \( t \) is the time. This means that the number of radionuclide nuclei produced increases with time.

### Step 2: Define the number of undecayed nuclei
Let \( n(t) \) represent the number of undecayed nuclei at time \( t \). The production of nuclei increases the number of undecayed nuclei, while decay decreases it.

### Step 3: Write the rate of decay
The rate of decay of the radionuclide is proportional to the number of undecayed nuclei present. According to the radioactive decay law, this can be expressed as:
\[ \text{Rate of decay} = -\lambda n(t) \]
where \( \lambda \) is the decay constant.

### Step 4: Write the net change in the number of undecayed nuclei
The net change in the number of undecayed nuclei over time can be expressed as:
\[ \frac{dn}{dt} = \text{Production rate} - \text{Decay rate} \]
Substituting the expressions we have:
\[ \frac{dn}{dt} = q_0 t - \lambda n(t) \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:
\[ \frac{dn}{dt} + \lambda n(t) = q_0 t \]

### Conclusion
The differential equation that correctly represents the process is:
\[ \frac{dn}{dt} + \lambda n(t) = q_0 t \]