Solving a Differential Equation

[Obraz35]

Homework Statement

<br /> \[P^{&#039;}(t)+(\lambda +\mu )P(t)=\lambda \]<br /> <br />

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
<br /> \[e^{(\lambda +\mu )t}\]<br />

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

Integrating both sides,
<br /> \[e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C\]<br />

which seems to give me
<br /> \[P(t)=\frac{\lambda}{\lambda + \mu}\]<br />

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

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]

Homework Statement

<br /> \[P^{&#039;}(t)+(\lambda +\mu )P(t)=\lambda \]<br /> <br />

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
<br /> \[e^{(\lambda +\mu )t}\]<br />

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

Integrating both sides,
<br /> \[e^{(\lambda + \mu)t}P(t)=\frac{\lambda e^{(\lambda + \mu)t}}{\lambda + \mu} + C\]<br />

You were doing great up to here (above). You multiplied both sides of the equation by
<br /> e^{-(\lambda + \mu)t}<br />
but forgot to multiply the constant C.

which seems to give me
<br /> \[P(t)=\frac{\lambda}{\lambda + \mu}\]<br />

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

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.