\frac{d}{dx}\left(\frac{dy}{dx}\right)=\:6x\:+\:3

\frac{d}{dx}\left(\frac{dy}{dx}\right)=\:6x\:+\:3 \\ 

 solve\; the\; equation\; using\; case\; 1,\\

let\; u=\frac{dy}{dx} \\
\int \:\frac{du}{dx}\:=\:\int \:\left(6x\:+\:3\right)\\

using\;  separation\; of\; variable\; divide\;   both\;  sides\;  by\;  dx,\\

\int \:du\:=\:\int \:\left(6x\:+\:3\right)dx\\

by\; integrating\; using\; the\; sum\; rule:\\
we\; get,\\

u=3x^2\:+\:3x\:+\:C_1\\

substitute\; the\; value\; of\; u=\frac{dy}{dx} \\

\frac{dy}{dx}=3x^2\:+\:3x\:+\:C_1\\

using\;  separation\; of\; variable:\\

\int \:dy = \int \:\left(3x^2+3x\:+\:C_1\right)dx\\

apply\; the\; sum\; rule:\\

\int \:dy=\int \:3x^2dx+\int  \:3xdx+ \int \:C_1dx\\

y=x^3+\frac{3x^2}{2}+C_1x+C_2\\