Question

If \(y=y(x)\) is the solution of the equation
\(e^{\sin y}\cos y\,\frac{dy}{dx}+e^{\sin y}\cos x=\cos x\), \(y(0)=0\); then
\(1+y\left(\frac{\pi}{6}\right)+\frac{\sqrt3}{2}\,y\left(\frac{\pi}{3}\right)+\frac{1}{\sqrt2}\,y\left(\frac{\pi}{4}\right)\) is equal to ____________.

Hint:

Put \(u=e^{\sin y}\) so that \(u'=e^{\sin y}\cos y\,y'\); the equation becomes linear in \(u\).

Answer 1

Correct Answer: 1

Let \(u=e^{\sin y}\). Then \(\frac{du}{dx}=e^{\sin y}\cos y\,\frac{dy}{dx}\), so the equation is \(\frac{du}{dx}+u\cos x=\cos x\).
Integrating factor \(=e^{\int\cos x\,dx}=e^{\sin x}\).
\(u\,e^{\sin x}=\int\cos x\,e^{\sin x}dx=e^{\sin x}+C\), so \(u=1+Ce^{-\sin x}\).
\(y(0)=0\Rightarrow u(0)=e^0=1\Rightarrow 1=1+C\Rightarrow C=0\).
So \(e^{\sin y}=1\), i.e. \(\sin y=0\); with \(y(0)=0\) and continuity, \(y(x)=0\) for all \(x\).
Expression \(=1+0+0+0=1\).
Answer: 1