Find the solution t the differential equation

Question

dz/dt = 2te^z that passes through the origin.

My answer is -In (1-t^2), but my answer is populating as incorrect. Can someone help please?

Answer 1

dz/dt = 2t e^z
e^-z dz = 2t dt
-e^-z = t^2 + c
e^-z = c-t^2
-z = ln(c-t^2)
z = -ln(c-t^2)
at(0,0) 0 = -ln(c)
c = 1

z = -ln(1-t^2)

I like your answer. Maybe they want

z = ln(1/(1-t^2))