The solution of the differential equation `(d theta)/(dt)=-k(theta-theta_(0))` where k is constant, is . . . .

Options

A. `theta=theta_(0)+ae^(-kt)`
B. `theta=theta_(0)+ae^(kt)`
C. `theta=2theta_(0)-ae^(kt)`
D. `theta=2theta_(0)-ae^(-kt)`

Explanation

To solve the differential equation \(\frac{d\theta}{dt} = -k(\theta - \theta_0)\), where \(k\) is a constant, we will follow these steps:

### Step 1: Rewrite the differential equation
We start with the given equation:
\[ \frac{d\theta}{dt} = -k(\theta - \theta_0) \]
This can be rearranged to:
\[ \frac{d\theta}{dt} + k\theta = k\theta_0 \]

### Step 2: Identify \(p\) and \(q\)
In the standard form of a linear first-order differential equation:
\[ \frac{dy}{dx} + p y = q \]
we can identify:
- \(p = k\)
- \(q = k\theta_0\)

### Step 3: Find the integrating factor
The integrating factor \(IF\) is given by:
\[ IF = e^{\int p \, dt} = e^{\int k \, dt} = e^{kt} \]

### Step 4: Multiply through by the integrating factor
We multiply the entire differential equation by the integrating factor:
\[ e^{kt} \frac{d\theta}{dt} + ke^{kt} \theta = ke^{kt} \theta_0 \]

### Step 5: Recognize the left-hand side as a derivative
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dt}(e^{kt} \theta) = ke^{kt} \theta_0 \]

### Step 6: Integrate both sides
Now we integrate both sides with respect to \(t\):
\[ \int \frac{d}{dt}(e^{kt} \theta) \, dt = \int ke^{kt} \theta_0 \, dt \]
This gives:
\[ e^{kt} \theta = \theta_0 e^{kt} + C \]
where \(C\) is the constant of integration.

### Step 7: Solve for \(\theta\)
Now, we can solve for \(\theta\):
\[ \theta = \theta_0 + Ce^{-kt} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \theta(t) = \theta_0 + Ce^{-kt} \]
where \(C\) is a constant determined by initial conditions.

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