Please Show Work 7 Consider the oneparameter family of difer

Please Show Work

7. Consider the one-parameter family of diferential equations defined by 7. Consider the one-parameter family of differential equations defined by dy =y2 +3y + a dt (15 points) a) Locate the bifurcation value. b) Draw at least three phase lines, one for the value of the parameter at the bifurcation value, and one each for values slightly larger and slightly smaller than the bifurcation value. c) Draw a bifurcation diagram for the family of equations.

Solution

Solution (a)

dy/dt = y^2 + 3^y + a

Given quadratic equation is in form of ay²+by+c and comparing will yield result as

a = 1, b = 3, c = a

Solution for the equation will be given as

y = -3 + sqrt(9 - 4a)/ 2

We see that y = 0 when a = 9/4 which is the bifurcation value.