If \(\cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y \sin x=6 x, 0 \lt x \lt \frac{\pi}{2}\), then general solution of the differential equation is

\(\begin{aligned}
& \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y \sin x=6 x \\
& \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}-(\tan x) y=6 x \sec x
\end{aligned}\)
This equation is of the form \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{p} y=\mathrm{Q}\)
\(\therefore \quad \text { If }=\mathrm{e}^{\int \mathrm{pdx}}=\mathrm{e}^{-\int \tan x d x}=\mathrm{e}^{\log x \cos x}=\cos x\)
\(\therefore \quad\) Solution of given equation is
\(\begin{aligned}
& y \cdot \cos x=\int 6 x \times \sec x \times \cos x \mathrm{~d} x+\mathrm{c} \\
& \Rightarrow y \cos x=\int 6 x \mathrm{~d} x+\mathrm{c} \\
& \Rightarrow y \cos x=3 x^2+\mathrm{c}
\end{aligned}\)