# Thread: [SOLVED] Differential Equation Help

1. ## [SOLVED] Differential Equation Help

Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

Here it is:
"find the equation of R(t) when

$\displaystyle \frac{dE}{dt}= -\gamma_{b}E$ where $\displaystyle E(0) = E_0$ and $\displaystyle \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,

$\displaystyle \frac{dS}{dt}= -\gamma_{s}S$ where $\displaystyle S(0) = S_0$ and $\displaystyle \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$

and

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $\displaystyle M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.

Now I know that

$\displaystyle S(t) = S_0 e^{-\gamma_{s} t}$

$\displaystyle E(t) = E_0 e^{-\gamma_{b} t}$

and then by substituting these values into $\displaystyle \frac{dR}{dt}$ we obtain

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$

this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks

2. Originally Posted by Maccaman
Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

Here it is:
"find the equation of R(t) when

$\displaystyle \frac{dE}{dt}= -\gamma_{b}E$ where $\displaystyle E(0) = E_0$ and $\displaystyle \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,

$\displaystyle \frac{dS}{dt}= -\gamma_{s}S$ where $\displaystyle S(0) = S_0$ and $\displaystyle \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$

and

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $\displaystyle M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.

Now I know that

$\displaystyle S(t) = S_0 e^{-\gamma_{s} t}$

$\displaystyle E(t) = E_0 e^{-\gamma_{b} t}$

and then by substituting these values into $\displaystyle \frac{dR}{dt}$ we obtain

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$

this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
Just integrate with respect to t.

3. Originally Posted by Maccaman
Hi all,
Earlier I started a thread with a past exam question that I wanted to understand before I attempted a homework question. It is similar to the question I posted earlier, but Im still having trouble with it.

Here it is:
"find the equation of R(t) when

$\displaystyle \frac{dE}{dt}= -\gamma_{b}E$ where $\displaystyle E(0) = E_0$ and $\displaystyle \gamma_b = \gamma_{b_1} + \gamma_{b_2} + \lambda_p$,

$\displaystyle \frac{dS}{dt}= -\gamma_{s}S$ where $\displaystyle S(0) = S_0$ and $\displaystyle \gamma_s = \gamma_{s_1} + \gamma_{s_2} + \lambda_p$

and

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E + \gamma_{s_1} MS$
where $\displaystyle M, \gamma_{b_1}, \gamma_{b_2}, \gamma_{b}, \gamma_{s_1}, \gamma_{s_2}, \gamma_{s}, \lambda_p$ are all constants.

Now I know that

$\displaystyle S(t) = S_0 e^{-\gamma_{s} t}$

$\displaystyle E(t) = E_0 e^{-\gamma_{b} t}$

and then by substituting these values into $\displaystyle \frac{dR}{dt}$ we obtain

$\displaystyle \frac{dR}{dt} = \gamma_{b_1} E_0 e^{-\gamma_b t} + \gamma_{s_1} M S_0 e^{-\gamma_s t}$

this is where I get lost. How do I continue from here? What type of differential equation is this? Thanks
Well, as the right hand side doesn't depend on R all you need to do is integrate both sides with respect to t.

-Dan

4. ahhhhhh, of course I feel so stupid now
thanks

5. Originally Posted by Maccaman
ahhhhhh, of course I feel so stupid now
thanks
There is nothing more difficult than seeing the obvious after you have done something complicated.

-Dan

6. Is this answer correct? (my integration rules are a bit dodgy)

$\displaystyle R(t) = - \frac{\gamma_{b_1} E_0 e^{-\gamma_b t}}{\gamma_b} - \frac{\gamma_{s_1} M S_0 e^{-\gamma_s t}}{\gamma_s}$

7. Originally Posted by Maccaman
Is this answer correct? (my integration rules are a bit dodgy)

$\displaystyle R(t) = - \frac{\gamma_{b_1} E_0 e^{-\gamma_b t}}{\gamma_b} - \frac{\gamma_{s_1} M S_0 e^{-\gamma_s t}}{\gamma_s}$
Looks good to me.

-Dan