Solving a Differential Equation

  • Thread starter Obraz35
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
35,393
7,271

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

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