# Solving a Differential Equation

## Homework Statement

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

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.

## The Attempt at a Solution

The first thing I did was multiply both sides by
$$$e^{(\lambda +\mu )t}$$$

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

Integrating both sides,
$$$e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C$$$

which seems to give me
$$$P(t)=\frac{\lambda}{\lambda + \mu}$$$

but I know that this is not correct since I am supposed to showing that the solution is
$$$P(t)=\frac{\lambda}{\lambda + \mu}(1 - e^{-(\lambda + \mu)t})+P(0)e^{-(\lambda + \mu)t}$$$.

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.

Mark44
Mentor

## Homework Statement

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

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.

## The Attempt at a Solution

The first thing I did was multiply both sides by
$$$e^{(\lambda +\mu )t}$$$

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

Integrating both sides,
$$$e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C$$$
You were doing great up to here (above). You multiplied both sides of the equation by
$$e^{-(\lambda + \mu)t}$$
but forgot to multiply the constant C.
which seems to give me
$$$P(t)=\frac{\lambda}{\lambda + \mu}$$$

but I know that this is not correct since I am supposed to showing that the solution is
$$$P(t)=\frac{\lambda}{\lambda + \mu}(1 - e^{-(\lambda + \mu)t})+P(0)e^{-(\lambda + \mu)t}$$$.

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.