Question

What is the solution of the differential equation \[\cot ydx = xdy\]?
A. \[y = \cos x\]
B. \[x = c\sec y\]
C. \[x = \sin y\]
D. \[y = \sin x\]

Answer 1

Hint: First we will separate the variables of the differential equation. Then we will integrate both sides of the differential equation to get the solution. By using the logarithm formula we will simplify the solution.

Formula Used: Integrating formula:
\[\int {\tan \theta d\theta } = \log \sec \theta + c\]
\[\int {\dfrac{1}{x}dx} = \log x + c\]
Product formula for logarithm:
\[\log a + \log b = \log ab\]


Complete step by step solution: Given differential equation is
\[\cot ydx = xdy\]
Separate the variables of the differential equation:
\[ \Rightarrow \dfrac{{dx}}{x} = \dfrac{{dy}}{{\cot y}}\]
\[ \Rightarrow \dfrac{{dx}}{x} = \tan ydy\]
Taking integration on both sides:
\[ \Rightarrow \int {\dfrac{{dx}}{x}} = \int {\tan ydy} \]
Applying integration formula:
\[ \Rightarrow \log x = \log \sec y + \log c\]
Applying the product rule of logarithm:
\[ \Rightarrow \log x = \log c\sec y\]
Applying the antilogarithm formula:
\[ \Rightarrow x = c\sec y\]

Option ‘B’ is correct

Additional Information: The constant that is added with the result of the integration is known as the integration constant. Usually, the integration constant denotes by c. Using antiderivative we are unable to get the actual expression. Thus we add an integration constant with the result of integration to get the family of solutions of the differential equation.
A solution of a differential equation is known as the general solution if the value of the constant is unknown.


Note: Students often do mistake to integrating \[\tan ydy\]. They used a wrong formula that is \[\int {\tan \theta d\theta } = \log \cos \theta + c\]. The correct formula is \[\int {\tan \theta d\theta } = \log \sec \theta + c\].