Question

Find the particular solution of the differential equation $ \dfrac{{dy}}{{dx}} + 2y\,\tan x = \sin x $ given that $ y = 0 $ when $ x = \dfrac{\pi }{3} $ .

Answer 1

Hint: Find the integration factor and use general form for differential equation solution.
First, we are going to compare the given equation with the general differential equation form and get the P and Q values, which are needed for finding the integrating factor and then use the solution form to get the solution of the differential equation and substitute given values and get the constant value and find the required solution.

Complete step by step solution:
We are given a differential equation
 $ \dfrac{{dy}}{{dx}} + 2y\,\tan x = \sin x $
The general differential equation of the form
 \[\dfrac{{dx}}{{dy}} + Py = Q\]
Now, on comparing both of them we will get
 \[P = 2tanx\] and \[Q = sinx\]
Now, we are going to find the integrating factor
 \[IF = {e^{\int P}}^{dx}\]
Now, we will substitute the value and get the factor
 \[
  IF = {e^{\int 2tanxdx}} \\
  IF = {e^{2logsecx}} \\
  IF = {e^{logse{c^2}x}} \\
  IF = (se{c^2}x) \;
\]
Now, using the general solution form, we will find the solution
 \[y(IF) = \int (Q \times IF)dx + c\]
Now, we are going to substitute the values into this general form.
 \[
\Rightarrow y(se{c^2}x) = \int sinx.se{c^2}xdx + c \\
\Rightarrow y.se{c^2}x = \int tanx.secxdx + c \\
\Rightarrow y.se{c^2}x = secx + c \;
\]
We have to make the variable free from radical terms. So,
 $\Rightarrow y = \dfrac{{\sec x}}{{{{\sec }^2}x}} + \dfrac{c}{{{{\sec }^2}x}} $
 $\Rightarrow y = \cos x + c\,\,{\cos ^2}x $ --------(1)
We are going to need to need to find the value of constant c
So, we substitute the values of x and y into the form
 $
  0 = \cos \dfrac{\pi }{3} + C\,{\cos ^2}\dfrac{\pi }{3} \\
  C{\left( {\dfrac{1}{2}} \right)^2} = \dfrac{{ - 1}}{2} \\
  C = \dfrac{{ - 4}}{2} \\
  C = - 2 \;
 $
Substitute this value into the equation (1). Then we get
 $ y = \cos x - 2{\cos ^2}x $
This is the required solution of the given differential equation.
So, the correct answer is “ $ y = \cos x - 2{\cos ^2}x $ ”.

Note: For solving this problem, we should be good with integration values as it helps in finding the integration factor and also, we should know the general form of the solution and general form of differential equation, as we get required value for solving.