Let y=y(x) solve dydx-|A|=0 for x0, where A=pmatrixy x&10&-1pmatrix (schematic 33 determi…

Question

Let $y=y(x)$ solve $\dfrac{dy}{dx}-|A|=0$ for $x>0$, where $A=\begin{pmatrix}y\sin x&1\0&-1\end{pmatrix}\dots$ (schematic $3	imes3$ determinant from source). If $y(\pi)=\pi+2$, find $y\left(\dfrac{\pi}{2}ight)$.

Accepted Answer

$|A|=-\dfrac yx+2\sin x+2$, giving $\dfrac{dy}{dx}+\dfrac{y}{x}=2\sin x+2$; I.F.$=x$.

$$xy = x^2-2x\cos x+2\sin x+C.$$

Using $y(\pi)=\pi+2$: $C=0$. At $x=\pi/2$: $y=\dfrac{\pi}{2}+\dfrac{4}{\pi}$.

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