Find the general solution of the differential equation: $\frac{dr}{d\theta }+r\sec \left(\theta \right)=\cos \left(\theta \right)$.

• A. $r=\cos \left(\theta \right)+C\sec \left(\theta \right)$
• B. $r=\sin \left(\theta \right)\tan \left(\theta \right)+C\csc \left(\theta \right)$
• C. $r=\sin \left(\theta \right)+C\cos \left(\theta \right)$
• D. $r=\cos \left(\theta \right)\tan \left(\theta \right)+C\sec \left(\theta \right)$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is a first-order linear differential equation of the form \( \frac{dr}{d\theta} + r \sec(\theta) = \cos(\theta) \). The goal is to find the general solution for \( r \).
2. Step 2: Rewrite the equation in standard form \( \frac{dr}{d\theta} + P(\theta)r = Q(\theta) \), where \( P(\theta) = \sec(\theta) \) and \( Q(\theta) = \cos(\theta) \).
3. Step 3: Find the integrating factor \( \mu(\theta) \), which is given by \( \mu(\theta) = e^{\int P(\theta) \, d\theta} \). Compute \( \int \sec(\theta) \, d\theta \), which results in \( \ln|\sec(\theta) + \tan(\theta)| \). Thus, \( \mu(\theta) = \sec(\theta) + \tan(\theta) \).
4. Step 4: Multiply through the differential equation by the integrating factor \( \mu(\theta) \). This transforms the equation into \( \frac{d}{d\theta} \left[ r \cdot \mu(\theta) \right] = Q(\theta) \cdot \mu(\theta) \). Substitute \( \mu(\theta) \) and \( Q(\theta) \) into the equation.
5. Step 5: Integrate both sides with respect to \( \theta \). The left-hand side simplifies to \( r \cdot \mu(\theta) \), and the right-hand side involves integrating \( \cos(\theta) \cdot (\sec(\theta) + \tan(\theta)) \). Solve for \( r \) by dividing through by \( \mu(\theta) \), and include the constant of integration \( C \). The general solution is \( r = \cos(\theta) \tan(\theta) + C \sec(\theta) \).