
Missing Solutions of DE
The question is about the initial value problem
$\displaystyle \frac{dy}{dt} = \frac{t \ cos \ 2t}{y}$
The question askes to find a oneparameter family of solutions to the DE, and then it asks: "Are there any solutions to the differential equation that are missing from the set of solutions that you found? Explain."
Attempt:
I've found the general solution by separation of variables:
$\displaystyle \int \ y \ dy = \int \ t \ cos 2t \ dt$
$\displaystyle \frac{y^2}{2} = \frac{1}{4} (2t \ sin (2t)+ cos(2t)) + c$
$\displaystyle \therefore y = \pm \sqrt{t \ sin (2t)+ \frac{1}{2} \ cos (2t) + k}$
So, how do I know whether there are any missing solutions or not? Any explanation would be greatly appreciated.