Q.Solve : $\frac{dy}{dx}+y\tan x=\cos^3 x$

Linear ODE: I.F. $=\sec x$; then $y\sec x=\int\cos^{2}x\,dx=\tfrac{x}{2}+\tfrac{\sin 2x}{4}+C$.

In the TN HSC Class-12 Business Maths differential-equations topic, an equation of the form $\dfrac{dy}{dx}+Py=Q$ is solved using the integrating factor $\text{I.F.}=e^{\int P\,dx}$.

Step 1 — identify $P$ and $Q$. Here $P=\tan x,\ Q=\cos^{3}x$.

Step 2 — integrating factor.

$\text{I.F.}=e^{\int \tan x\,dx}=e^{\log \sec x}=\sec x.$

Step 3 — general solution formula $y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx$:

$y\sec x=\int \cos^{3}x\cdot\sec x\,dx=\int \cos^{2}x\,dx.$

Step 4 — integrate $\cos^{2}x$. …