Question

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is ______.

• A. $\cos x = c \csc y$
• B. $\sin x = c \sec y$
• C. $\sin x = c \cos y$
• D. $\cos x = c \sin y$

Hint:

Using $\int \tan \theta \,d\theta = -\ln|\cos \theta|$ is generally much safer than using $\ln|\sec \theta|$ during separation of variables, as it avoids complex fraction inversions later when removing the logarithms!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a first-order differential equation. Because it consists entirely of products, it can be easily solved using the "separation of variables" method.

Step 2: Detailed Explanation:
The given differential equation is:
$\frac{dy}{dx} = \cot x \cdot \cot y$
Separate the variables by bringing all $y$ terms to the left side and all $x$ terms to the right side:
$\frac{dy}{\cot y} = \cot x \, dx$
Using the reciprocal identity $\frac{1}{\cot y} = \tan y$:
$\tan y \, dy = \cot x \, dx$
Integrate both sides:
$\int \tan y \, dy = \int \cot x \, dx$
The standard integration formulas are:
$\int \tan y \, dy = -\ln|\cos y|$ (or $\ln|\sec y|$)
$\int \cot x \, dx = \ln|\sin x|$
Using the negative cosine version for the left side:
$-\ln|\cos y| = \ln|\sin x| + C_1$
To make manipulation easier, let the constant of integration be expressed as a logarithm, $C_1 = \ln C$:
$-\ln|\cos y| = \ln|\sin x| + \ln C$
Bring the logarithmic terms together:
$\ln|\sin x| + \ln|\cos y| = -\ln C$
Use the logarithm addition property ($\ln A + \ln B = \ln(AB)$):
$\ln|\sin x \cdot \cos y| = \ln(C')$ (where $C' = e^{-\ln C}$ is just a new arbitrary constant)
Remove the natural logarithm from both sides:
$\sin x \cdot \cos y = C'$
Now, rearrange the equation to match the provided multiple-choice options. Let's isolate $\sin x$:
$\sin x = \frac{C'}{\cos y}$
Using the reciprocal identity $\frac{1}{\cos y} = \sec y$:
$\sin x = C' \sec y$
Letting the arbitrary constant be denoted simply as $c$:
$\sin x = c \sec y$

Step 3: Final Answer:
The general solution is $\sin x = c \sec y$, matching option (b).