- #1

- 31

- 0

## Homework Statement

[tex]

\[P^{'}(t)+(\lambda +\mu )P(t)=\lambda \]

[/tex]

I have never worked with differential equations before and I am trying to work off of the one example we did in class, but I can't figure out where I am going wrong.

## Homework Equations

## The Attempt at a Solution

The first thing I did was multiply both sides by

[tex]

\[e^{(\lambda +\mu )t}\]

[/tex]

Then,

[tex]

\[\frac{d}{dt}[e^{(\lambda + \mu)t}P(t)]=\lambda e^{(\lambda + \mu)t}\]

[/tex]

Integrating both sides,

[tex]

\[e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C\]

[/tex]

which seems to give me

[tex]

\[P(t)=\frac{\lambda}{\lambda + \mu}\]

[/tex]

but I know that this is not correct since I am supposed to showing that the solution is

[tex]

\[P(t)=\frac{\lambda}{\lambda + \mu}(1 - e^{-(\lambda + \mu)t})+P(0)e^{-(\lambda + \mu)t}\]

[/tex].

I don't think I am solving for C correctly but since I have never really been taught this I'm not quite sure what to do or how to get that solution. I'd really appreciate it if someone could let me know where I am going wrong.