**Hello, I'm a little stuck on this problem:**

**Find a solution to the differential equation subject to the initial conditions.**

dz

dt = te^{z} , through the origin.

**Here is how I tried to solve the problem:**

__dz__

dt = te^{z}

__multiply both sides by dt, and then divide both sides by e__^{z }to get like variables with like variables:

__dz__

e^{z = tdt }

*Take the integral of both sides:*
(integral sign) 1/e

^{z} dz = (integral sign) t dt

*substitution*
w = e

^{z}
dw = e

^{z} dz

(integral sign)1/w dz = (integral sign) t dt

ln|w| = t

^{2}/2 + C

ln|e^{z}| = t^{2}/2 + C

__exponentiate both sides to get rid of the natural log:__

e^{ln|ez|} = e^{t2/2 + C the e and the natural log cancel out }e^{z} = e^{(}^{t^2)}e^{c }e^{z} = Ce^{(t^2) ----(where C is just a constant)}

**Now this is where I am stuck. Have I done it correctly so far? What do I need to do next? Please help. Thank you.**