Question

Solve the differential equation. (Use C for any needed constant.)

Answer 1

\[\frac{ dz }{ dt }+4e^{t+z}=0\]

Answer 2

\[\frac{ dz }{ dt }=-4e^{t+z}\]\[\frac{ dz }{ dt }=-4e^te^z\]

I am not sure how to set up the next step.

Answer 3

What you've done so far is good. What we wanna do is basically think of \( dz\) and \(dt\) as things we can multiply and divide right now, and so you want to rearrange this equation so that the left hand side has only 'z' in it and the right hand side has only 't' in it. Give it a shot, try something out and I'll help you figure out the rest.

Answer 4

so, would I divide \[e^z \] to the left side?

Answer 5

\[\frac{ 1 }{ e^z }\frac{ dz }{ dt }= - 4e^t\]\[\int\limits_{}^{}e^{-z}dz=\int\limits_{}^{}-4e^tdt\]

Like that?

Answer 6

Yeah, looking good, keep going

Answer 7

Okay, I got it now, \[z = \ln(\frac{ 1 }{ 4e^t-C })\]

Thanks! :)