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.