91Ó°ÊÓ

Anexplicit solutionis one in which the dependent variable is expresseddirectlyin terms of the independent variable and constants.

Let the given function be\(y = {(1 - sinx)^{ - {\textstyle{1 \over 2}}}}\).

Then, the first derivative of the function is,

\(y' = \frac{{cosx}}{2}{(1 - sinx)^{ - {\textstyle{3 \over 2}}}}\)

Substitute\(y\)and \(y'\) into the left-hand side of the differential equation.

\(\begin{aligned}{c}2\left( {\frac{{cosx}}{2}{{(1 - sinx)}^{ - \frac{2}{2}}}} \right) = {(1 - sinx)^{ - \frac{\pi }{2}}}cosx\\{(1 - sinx)^{ - \frac{3}{2}}}cosx = {(1 - sinx)^{ - \frac{3}{2}}}cosx\end{aligned}\)

That is same as the right-hand side of the differential equation. The indicated function is an explicit solution of the given differential equation.

Hence the domain of the solution while considering the solution as a function is,

\(\begin{aligned}{l}y = \frac{1}{{{{(1 - sinx)}^{1/2}}}}\\1 - sinx > 0\\sinx < 1\\ - \frac{{3\pi }}{2} + 2\pi n < x < \frac{\pi }{2} + 2\pi n\\D:x \in \left( { - \frac{{3\pi }}{2} + 2\pi n,\frac{\pi }{2} + 2\pi n} \right),n = 1,2,3, \ldots \end{aligned}\)

And the interval is \(I:\left( { - \frac{{3\pi }}{2} + ,\frac{\pi }{2}} \right)\).