$($a$)$ Show that

y squared plus x minus $3$ equals $0$y$2+$x$3=0$

is an implicit solution to

StartFraction dy Over dx EndFraction equals negative StartFraction $1$ Over $2$ y EndFractiondydx$=12$y

on the interval

left parenthesis negative infinity comma $3$ right parenthesis$($$,3).$

$($b$)$ Show that

xy Superscript $7$ Baseline minus xy Superscript $7$ Baseline cosine x equals $1$xy$7$xy$7$cosx$=1$

is an implicit solution to the differential equation

StartFraction dy Over dx EndFraction equals StartFraction left parenthesis negative x sine x plus cosine x minus $1$ right parenthesis y Over $7$ left parenthesis x minus x cosine x right parenthesis EndFractiondydx$=($xsinx$+$cosx$1)$y$7($x$$xcosx$)$

on the interval

left parenthesis $0$ comma $2$ pi right parenthesis$(0,2$$).$