Solve the differential equation

Question

Solve the differential equation

dz/dt +5e^t+z = 0

Please show detailed work. I know that the answer for sure is not z= -t+c

Answer 1

e^t+z = e^te^z

So, dz/dt = -5e^te^z

Separate variables to get e^-zdz = -5e^tdt

Integrate both sides: -e^-z = -5e^t + C

e^-z = 5e^t + C

-z = ln(5e^t + C)

z = -ln(5e^t + C)

Answer 2

Subtract the exponential from both sides to get:

dz/dt = -5e^t+z

Now, multiply both sides by e^-z dt to get:

e^-z dz = -5e^tdt

Integrating both sides, we have:

-e^-z = -5e^t + C

e^-z = 5e^t + C (Negative arbitrary constant is still arbitrary, so can omit negative sign).

-z = ln(5e^t + C)

z = -ln(5e^t + C)