Question

The solution of the equation $\frac{dy}{dx}+ytanx=x^{m}cosx$ is

• A. $(m+1)y=x^{m+1}cosx+c(m+1)cosx$
• B. $(my=(x^m+c)cosx$
• C. $y=(x^{m+1}+c)cosx$
• D. $Noneofthese$

Answer 1

The correct option is A $(m+1)y=x^{m+1}cosx+c(m+1)cosx$

This is the linear equation of the form
$\frac{dy}{dx}+Py=Q,$
Where P = tan x and $Q=x^{m}cosx$
Now integrating factor $(I.F.)=e^{\int pdx}=e^{\int tandx}{e}^{logsecx=secx}$
Thus solution is given
by, $y.e^{\int pdx}=\int Q.e^{\int Pdx}dx+c\Rightarrow y.secx=\int x^m.cosxsecxdx+c\Rightarrow ysecx=\frac{x^{m+1}}{m+1}+c\Rightarrow (m+1)y=x^{(m+1)}cosx+c(m+1)cosx.$