Without completing the square, make a guess at identifying each of the following conic sections:

$($a$)y=x^2-2y^2+1,$

$($b$)x^2+y^2-2=0,$

$($c$)y+2x=x^2+3,$

$($d$)x^2-2y^2=4+y,$

$($e$)y^2+3x=12.$