Use separation of variables to provide implicit solutions to each differential equation. If an initial condition is provided, find the solution satisfying it$.$

$($a$)\frac{dy}{dx}=\frac{3x^2}{3y^2-4},y(1)=0$

$($d$)\frac{dy}{dt}=si{n}^2(t)y^2$

$($b$)\frac{dy}{dx}=\frac{y^2}{x},y(1)=2$

$($e$)\frac{dy}{dx}=\frac{x^2}{y}{e}^{2x3y}.$

$($c$)\frac{dy}{dt}=\frac{t^2{e}^t}{y\sqrt[\phantom{2}]{1y^2}}$