Question

Find the solution of the differential equation: $\frac{dy}{dx}+y\tan x=\sec x$.

Answer 1

Given differential equation is: $\frac{dy}{dx}+y\tan x=\sec x$

Comparing it with $\frac{dy}{dx}+Py=Q$, we get

$P=\tan x$ and $Q=\sec x$

$I.F.=e^{\int Pdx}$

$=e^{\int \tan xdx}$

$=e^{\log \sec x}$

$=\sec x$

Hence the required solution is $y\cdot e^{\int Pdx}=\int Q\cdot e^{\int Pdx}\cdot dx$

$\Rightarrow y\cdot \sec x=\int \sec x.\sec xdx+C$

$y\cdot \sec x=\int {\sec }^{2}xdx+C$

$y\cdot \sec x=\tan x+C$

$\Rightarrow y\sec x=\tan x+C$

$\Rightarrow y=\sin x+C\cos x$.