Locate the bifurcation values of a for the one-parameter family and describe the bifurcation that takes place at each such value.

dy/dt=e^(-y^2)+a
I got that the equilibrium points will be at -sqrt(ln(a)). I don't know if that is fine. I took the partial derivative with respect to y and got -2ye^(-y^2) and that is zero when a is 0 if I plug in -sqrt(ln(a)) as y.

To determine the bifurcation values of a for the given one-parameter family of DE, let's start by finding the equilibrium points.

Given that the differential equation is:
dy/dt = e^(-y^2) + a

To find the equilibrium points, we set dy/dt equal to 0:
0 = e^(-y^2) + a

Now, we can isolate the exponential term:
e^(-y^2) = -a

Next, take the natural logarithm of both sides to solve for y:
ln(e^(-y^2)) = ln(-a)
-y^2 = ln(-a)

Let's solve for y:
y^2 = -ln(a)

Taking the square root of both sides, we obtain:
y = ±sqrt(-ln(a))

So, the equilibrium points are at y = sqrt(-ln(a)) and y = -sqrt(-ln(a)).

Now, let's analyze the bifurcation that takes place at each value of a:

1. When a > 0:
For positive values of a, the equilibrium points exist when -ln(a) is negative, which means that a is between 0 and 1. In this case, there are two real equilibrium points at y = sqrt(-ln(a)) and y = -sqrt(-ln(a)). These equilibrium points are stable, and the solutions to the differential equation will approach them over time.

2. When a = 0:
When a equals 0, the equation becomes:
dy/dt = e^(-y^2) + 0
dy/dt = e^(-y^2)

In this case, there are no real equilibrium points. The solutions to the differential equation will continue to evolve without converging to any fixed values.

3. When a < 0:
For negative values of a, the equilibrium points exist when -ln(a) is positive, which means that a is less than 0. In this case, there are again no real equilibrium points. The solutions to the differential equation will continue to evolve without converging to any fixed values.

Thus, the bifurcation occurs when a goes from positive to non-positive (a = 0), causing a change in the number and stability of the equilibrium points.