Question

Solve:
$\dfrac{{dy}}{{dx}} = \cot x \times \cot y$

Answer 1

This question is based on the variable separation method of solving differential equations. First we will separate variable to their respective differential variable then we will use the standard formula of integration to find the integration of the given differential equation.
Formula used :
$\int {\tan \theta } = \int {\dfrac{{\sin \theta }}{{\cos \theta }}dy} = \ln |\sec \theta | + c$
$\int {\cot \theta } = \int {\dfrac{{\cos \theta }}{{\sin \theta }}dy} = \ln |\sin \theta | + c$
In every integration, c has to be added. C is referred to as a constant.

Complete step by step answer:
Derivative of the function can be simply defined as the Slope of the curve. If the above example is considered, it cannot be solved by differentiation. Integration can make this sum easy.
To find: $\dfrac{{dy}}{{dx}} = \cot x \times \cot y$
We will solve this sum in a little different manner, just to keep it simple.
We are going to make certain adjustments and then solve it by integration.
Let us divide both sides by cot y, and multiply both sides by dx, after doing it, we get,
$\dfrac{{dy}}{{\cot y}} = \cot x \times dx$ --- 1
We do this previous step as we get the variables in their right place, i.e. y is differentiated with respect to y on one side and x is differentiated with respect to x on the other side.
We know that, $\dfrac{1}{{\cot y}} = \tan y$
Therefore, equation 1 becomes,
$\tan ydy = \cot x \times dx$
Integrating on both sides, we get,
$\int {\tan ydy} = \int {\cot x \times dx}$
On solving the integration of both the sides, we get
$\ln |\sec y| = \ln |\sin x| + \ln c$
Taking ln|sinx| to the other side, we get,
$\ln |\sec y| - \ln |\sin x| = \ln c$
We know that, $\ln a - \ln b = \ln \dfrac{a}{b}$
Therefore, $\ln \dfrac{{\sec y}}{{\sin x}} = \ln c$
Taking antilog on both sides, we get,
$\dfrac{{\sec y}}{{\sin x}} = c$
Multiplying both the sides by sin x , we get,
$\sec y = c \times \sin x$
Therefore, $\dfrac{{dy}}{{dx}} = \cot x \times \cot y$ is $\sec y = c \times \sin x$

Note:
Integrals of standard functions must be known, making you solve the sum easily and quickly. Also, substitutions should be done properly. Mostly try to solve this type of complex functions by substitutions. Concepts of trigonometry should also be well known. Adding c on the right hand side is a compulsion while solving integration sums.