Let y=y(x) be the solution of dy/dx+ytan x=sin 2x on (-(π)/(2),(π)/(2)) with y(0)=1. Find y ((π)/(4)).

This is a linear equation with integrating factor $e^{\int\tan x\,dx}=\sec x$ on the given interval. Multiplying through, $\frac{d}{dx}(y\sec x)=\sin2x\sec x=2\sin x$, because $\sin2x=2\sin x\cos x$. Integrating gives $y\sec x=-2\cos x+C$, so $y=C\cos x-2\cos^2x$. The condition $y(0)=1$ gives $C-2=1$, hence $C=3$ and $y=3\cos x-2\cos^2x$. At $x=\pi/4$, $\cos x=\frac1{\sqrt2}$, so $y=\frac3{\sqrt2}-2\cdot\frac12=\frac{3}{\sqrt2}-1\approx1.1213$, i.e. $1.12$. Substituting the solution back into the equation verifies it, and a symbolic ODE solver returns the same function. The most common error is a sign mistake in the integrating factor or forgetting that sin 2x carries a factor cos x that cancels the secant, which would leave a much harder integral. The final answer is rounded to two decimal places as the question requires.