$($a$)$

What can you say about a solution of the equation

y$=$

$14$

$$y$2$

just by looking at the differential equation?

The function y must be decreasing $($or equal to $0)$ on any interval on which it is defined.The function y must be strictly decreasing on any interval on which it is defined.$$The function y must be strictly increasing on any interval on which it is defined.The function y must be equal to $0$ on any interval on which it is defined.The function y must be increasing $($or equal to $0)$ on any interval on which it is defined.

$($b$)$

Verify that all members of the family

y $=$

$4($x $+$ C$)$

are solutions of the equation in part $($a$).$

We substitute the values of y and y$$ and test the solution to see if the left hand side $($LHS$)$ is equal to the right hand side $($RHS$).$

y$=$

$4$x $+$ C

$$ y$=$

$($x $+$ C$)2$

$$

LHS$=$y$=$

$($x $+$ C$)2$

$=$

$14$

$$

x $+$ C

$2$

$=$

$14$

$$y$2=$ RHS

$($c$)$

Can you think of a solution of the differential equation

y$=$

$14$

$$y$2$

that is not a member of the family in part $($b$)?$

y $=0$ is a solution of y$=$

$14$

$$y$2$ that is not a member of the family in part $($b$).$

Every solution of

y$=$

$14$

$$y$2$ is a member of the family in part $($b$).$

$$

y $=$ e$4$x is a solution of y$=$

$14$

$$y$2$ that is not a member of the family in part $($b$).$

y $=$ x is a solution of y$=$

$14$

$$y$2$ that is not a member of the family in part $($b$).$

y $=4$ is a solution of y$=$

$14$

$$y$2$ that is not a member of the family in part $($b$).$

$($d$)$

Find a solution of the initial$-$value problem.

y$=$

$14$

$$y$2$ y$(0)=0\mathrm{.}25$

y $=$