By using a suitable substitution, bring the equation y'sin…

By using a suitable substitution, bring the equation $y'\sin x=y\ln y$ to a form with separated variables, and then integrate it, given $y(\pi/2)=1$.

Step 1: Separate variables

\[\frac{dy}{y\ln y}=\frac{dx}{\sin x},\qquad y>0,\ y\neq1,\ \sin x\neq0.\]

Step 2: Integrate

With $u=\ln y$, the left side integrates to $\ln|\ln y|$; the right side is the standard integral $\displaystyle\int\csc x\,dx=\ln\left|\tan\tfrac{x}{2}\right|$. Hence $\ln|\ln y|=\ln\left|\tan\tfrac{x}{2}\right|+C$, so

\[y=e^{C\tan(x/2)}.\]

Step 3: Apply the initial condition

With $y(\pi/2)=1$ and $\tan(\pi/4)=1$: $1=e^{C}$, so $C=0$, giving the particular solution

\[\boxed{\,y=1.}\]