Question

Solve the given differential equation.

$\frac{dy}{dx}-y\tan x=e^x$

• A. $I=\int e^x\sin xdx$
• B. $I=\int e^x\cos xdx$
• C. $I=-\int e^x\cos xdx$
• D. None of these.

Answer 1

Answer: (B)

$I=\int e^x\cos xdx$

$\frac{dy}{dx}-y\tan x=e^x$ ...(1)

Here $P=-\tan x\Rightarrow \int Pdx=-\int \tan xdx=\log \left(\cos x\right)$

$\therefore I.F.=e^{\int Pdx}=e^{\log \left(\cos x\right)}=\cos x$

Multiplying (1) by $I.F.$ we get

$\cos x\frac{dy}{dx}-y\sin x=e^x\cos x$

Integrating both sides, we get

$y\cos x=\int e^x\cos xdx+c\Rightarrow y\cos x=\frac{1}{2}{e}^x(\sin x+\cos x)+c$