Use the problem below

Discuss, and illustrate with examples, how to solve differential equations of the forms

dy$/$dx $=$ f$($x$)$

and

d$2$y$/$dx$2=$ f$($x$).$

and $(2)$ and $(3)$ of this section.

Solve:$$

dydx

$=$ f$($x$,$ y$)(2)$Subject to:y$($x$0)=$ y$0$

Solve:$$

d$2$ydx$2$

$=$ f$($x$,$ y$,$ y$\text{'})(3)$Subject to:y$($x$0)=$ y$0,$ y$\text{'}($x$0)=$ y$1$

Find a function whose second derivative is

y$\text{'}\text{'}=12$x $2$

at each point

$($x$,$ y$)$

on its graph and

y $=$x $+6$

is tangent to the graph at the point corresponding to

x $=1.$