Question

\[\cos y\frac{dy}{dx} = e^x , y\left( 0 \right) = \frac{\pi}{2}\]

Answer 1

\[\cos y\frac{dy}{dx} = e^x , y\left( 0 \right) = \frac{\pi}{2}\]
\[ \Rightarrow \cos y\ dy = e^x dx\]
Integrating both sides, we get
\[\int\cos y\ dy = \int e^x dx\]
\[ \Rightarrow \sin y = e^x + C . . . . . (1)\]
\[\text{ We know that at }x = 0, y = \frac{\pi}{2} . \]
Substituting the values of x and y in (1), we get
\[1 = 1 + C\]
\[ \Rightarrow C = 0\]
Substituting the value of C in (1), we get
\[\sin y = e^x \]
\[ \Rightarrow y = \sin^{- 1} \left( e^x \right)\]
\[\text{ Hence, }y = \sin^{- 1} \left( e^x \right)\text{ is the required solution }.\]