Question

what is the eq. of the curve passing through the point (π/2,1) and having slope sinx/x^2 - 2y/x at each point (x,y) with x≠0.

Answer 1

We need to use the solution methods of the first order differential equations for this. This particular case, is easier and solvable also this way. 

$Slope=\frac{dy}{dx}=\frac{sin\ x}{x^2}-\frac{2y}{x}\\ \\\frac{dy}{dx}+ (\frac{2}{x})y=\frac{sin\ x}{x^2}\\ \\x^2\frac{dy}{dx}+ (2x)y=sin\ x\\ \\L H S\ is\ derivative\ of\ x^2y\ wrt\ x.\\ \\\frac{d(x^2y)}{dx}= sin\ x\\ \\x^2y= \int\limits^{}_{} {Sin\ x} \, dx = - Cos\ x + C\\ \\y = \frac{C-Cos\ x}{x^2}\\ \\(\frac{ \pi }{2},1)\ \ is\ on\ the \ curve.\\ \\1=\frac{C-0}{ \pi^2/4 }\\ \\C = \frac{4}{ \pi^2}\\ \\The\ desired\ Curve\ is\ y = \frac{\frac{4}{ \pi^2 }-Cos\ x}{x^2},\ \ x \neq 0$