Consider the differential equation dy dt

= 2t y 2 + y 2

.

a. Find a one-parameter family of solutions to the DE (i.e., a formula for solutions, with one arbitrary constant in the formula).

b. Check that you have the correct answer in

(a) by substituting your answer back into the DE.

c. Find a solution to the DE that satisfies the initial condition y(0) =

1 2

.

d. For which range of t does the solution you found in part (c)

exist? (Don’t forget to include both positive and negative t.)

e. Calculate the solution to the DE with y(1) =

1 2

? For what range of t does this solution exist?

f. Calculate the solution to the DE with y(−3) = −

1 4

? For what range of t does this solution exist?