Differential Equations Question 164
Question: The solution of the equation $ \frac{dy}{dx}+y\tan x=x^{m}\cos x $ is
Options:
A) $ (m+1)y={x^{m+1}}\cos x+c(m+1)\cos x $
B) $ my=(x^{m}+c)\cos x $
C) $ y=({x^{m+1}}+c)\cos x $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
This is the linear equation of the form $ \frac{dy}{dx}+Py=Q $ , where $ P=\tan x $ and $ Q=x^{m}\cos x $
Now integrating factor (I.F.) $ ={e^{\int{Pdx}}}={e^{\int{\tan dx}}} $
$ ={e^{\log \sec x}}=\sec x $
Thus solution is given by, $ y.{e^{\int{Pdx}}}=\int{Q}.{e^{\int{Pdx}}}dx+c $
Therefore $ y.\sec x=\int{x^{m}}.\cos x.\sec xdx+c $
Therefore $ y\sec x=\frac{{x^{m+1}}}{m+1}+c $
Therefore $ (m+1)y={x^{m+1}}\cos x+c(m+1)\cos x $ .
BETA
