variable separable dr/dθ = r^2/θ; r(1) = 2 . the answer must be: lnθ+ 1/r = 1/2

Let's solve the differential equation \(\frac{dr}{d\theta} = \frac{r^2}{\theta}\) with the initial condition \(r(1) = 2\), and show that the solution is \(\ln\theta + \frac{1}{r} = \frac{1}{2}\).

### Step 1: Separate the Variables
Given:
\[
\frac{dr}{d\theta} = \frac{r^2}{\theta}
\]

Separate the variables by dividing both sides by \(r^2\) and multiplying both sides by \(d\theta\):
\[
\frac{1}{r^2} dr = \frac{1}{\theta} d\theta
\]

### Step 2: Integrate Both Sides
Integrate both sides with respect to their respective variables:
\[
\int \frac{1}{r^2} dr = \int \frac{1}{\theta} d\theta
\]

The integrals are:
\[
\int \frac{1}{r^2} dr = -\frac{1}{r} + C_1
\]
\[
\int \frac{1}{\theta} d\theta = \ln|\theta| + C_2
\]

So, we have:
\[
-\frac{1}{r} = \ln|\theta| + C
\]

Where \(C = C_2 - C_1\) is the constant of integration.

### Step 3: Solve for \(\frac{1}{r}\)
Rearrange the equation to solve for \(\frac{1}{r}\):
\[
\frac{1}{r} = -\ln|\theta| - C
\]

Multiply both sides by \(-1\) to match the desired form:
\[
\frac{1}{r} = C - \ln|\theta|
\]

### Step 4: Determine the Constant \(C\) using Initial Condition
Use the initial condition \(r(1) = 2\):
\[
\frac{1}{2} = C - \ln(1)
\]
Since \(\ln(1) = 0\), we get:
\[
\frac{1}{2} = C
\]

So, \(C = \frac{1}{2}\).

### Step 5: Write the Final Solution
Substituting \(C = \frac{1}{2}\) back into the equation, we get:
\[
\frac{1}{r} = \frac{1}{2} - \ln|\theta|
\]

Rearrange it to the desired form:
\[
\ln|\theta| + \frac{1}{r} = \frac{1}{2}
\]

Given that \(\theta > 0\), the absolute value can be removed:
\[
\ln\theta + \frac{1}{r} = \frac{1}{2}
\]

This confirms that the solution is correct.

Would you like more details or have any questions?

Here are five questions that build on this information:
1. How does the process change if the initial condition \(r(1) = 3\) instead of \(r(1) = 2\)?
2. Can you explain why the constant \(C\) was negative in this case?
3. How does this method apply to solving higher-order differential equations?
4. What happens if the differential equation is not separable? How would you approach it?
5. Can this method be extended to solve partial differential equations?

**Tip:** Always double-check your integration and substitution steps, especially when solving differential equations, as small errors can lead to incorrect results.

Learn how to solve the separable differential equation dr/dθ = r²/θ with the initial condition r(1) = 2. Detailed step-by-step solution including separation of variables, integration, solving for constants, and verifying the solution. Suitable for advanced undergraduate students and those studying differential equations.

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