Consider the differential equation

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

$($a$)$ Solve the differential equation by separating the variables, and find a general solution.

$($b$)$ Using Desmos, graph the direction field for this differential equation, along with several possible solution curves for different values of $c.$

$($c$)$ When you separated the variables in this equation, you assumed that $0$ on some interval. Now suppose that assumption fails, and $=0$ everywhere.

Check that this condition also describes a solution to your differential equation above $-$ a solution that we missed before.

$($d$)$ Use Desmos to plot this new constant solution as well as your previous solution curves and direction field. Does your graph make sense?

$($e$)$ Read $($or review$)$ the book's Example $2\mathrm{.}2\mathrm{.}5.$ With your group, talk through the process used in the book for solving this problem. How do you find the constant solutions for this differential equation? How many constant solutions are there, and why? What do these constant solutions look like on the direction field?

Please provide the graphs from Desmos so I can fully understand how to graph these equations.