Find the particular solution of the differential equation given that <img alt="y= 0" src=

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

Answers (1)

given D.E is $\frac{dy}{dx}+\left ( 2\tan x \right )y= \sin x---(i)$
which is a linear differential equation of the form
$\frac{dy}{dx}+py=Q$
where $p= 2\tan x,Q = \sin x$ , Integrating factor
$\because IF= e^{\int pdx}= e^{\int2\tan dx}= e^{2\log \sec x}= \sec ^{2}x$
The solution of equation (i) is given by
$y\cdot IF= \int Q \cdot IF\; dx+c$
$\Rightarrow y\sec ^{2}x= \int \sin x\cdot \sec ^{2}x\, dx+c$
$\Rightarrow y\sec ^{2}x= \int \sec x\tan\, x \, dx+c$
$\Rightarrow y\sec ^{2}x= \sec x+c$
put y = 0 when $x= \frac{\pi }{3}$
$\theta \cdot \sec ^{2}\frac{\pi }{3}= \sec \frac{\pi }{3}+c$
$0= 2+c$
$c= -2$ hence the perticular solution is
$y\sec ^{2}x= \sec x-2\; \; or\; \; y= \cos x-2\cos ^{2}x$