Question

x cos x $\frac{dy}{dx}$ + y (x sin x + cos x) = 1

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ x\cos x\frac{dy}{dx}+y\left(x\sin x+\cos x\right)=1 \\ \Rightarrow \frac{dy}{dx}+\left(\tan x+\frac{1}{x}\right)y=\frac{1}{x\cos x} \\ \mathrm{Comparing}\mathrm{with}\frac{dy}{dx}+Px=Q,\mathrm{we}\mathrm{get} \\ P=\tan x+\frac{1}{x} \\ Q=\frac{1}{x\cos x} \\ \mathrm{Now}, \\ \mathrm{I}.\mathrm{F}.=e^{\int \left(\tan x+\frac{1}{x}\right)dx}=e^{\log \left|x\sec x\right|}=x\sec x \\ \mathrm{So},\mathrm{the}\mathrm{solution}\mathrm{is}\mathrm{given}\mathrm{by} \\ xy\sec x=\int {\sec }^{2}xdx+C \\ \Rightarrow xy\sec x=\tan x+C \end{gathered}$$