Question

Solve y = p tan p + log (cos p), where, =

Answer 1

Given

y=p\tan p+\log (\cos p)

Differentiating the above w.r.t. x

\frac{dy}{dx}=p\sec^p\frac{dp}{dx}+\tan p\frac{dp}{dx}+\frac{1}{\cos p}\times(-\sin p)\frac{dp}{dx}

\implies p=p\sec^p\frac{dp}{dx}+\tan p\frac{dp}{dx}-\tan p\frac{dp}{dx}

\implies p=p\sec^2p\frac{dp}{dx}

\implies \sec^2p dp=dx

\implies\int \sec^2p dp=\int dx

\implies \tan p=x+c where c is a constant

\implies x=\tan p-c

Thus,

y=p\tan p+\log (\cos p) and x=\tan p-c is parametric solution of the equation, where p is a parameter...................