# First-order homogenous linear ODE in 2 functions, with information about initial conditions

What can be done about the following differential equation:

$$\frac{dP}{dt} + \frac{dR}{dt} +bR = 0$$ with: $$t\geq0\\ R(0) = 0 \\ P(k) + R(k)=\frac{1}{2}P(0)$$

Any suggestions? Thank you.

This is a simplification of https://math.stackexchange.com/questions/795232/saying-something-about-model-parameters-knowing-the-half-life-of-total-protein. It was obtained by adding eqs. 1 & 2, so that the first two terms in the RHS cancel.

• Hi. Can you show a bit of your work on this so far? This way it is easier to help you! Also any reference would be great. – MattAllegro May 15 '14 at 22:42
• Unfortunately it doesn't seem that I can separate the variables, so I actually have no idea what to do. I have started reviewing the theory on DEs. I have obtained the equation by adding eqs. 1 and 2 from math.stackexchange.com/questions/795232/… . – WindChimes May 15 '14 at 22:48
• Technically $P = k$ and $R = c_1 e^{-bt}$ is a solution. – IAmNoOne May 15 '14 at 22:50
• There are infinitely many solutions because you have more unknowns than equations. – Tunococ May 15 '14 at 23:09

Set $$\frac{dP}{dt} = -bR.$$

$$\frac{dR}{dt} = 0$$

The solution is $R = k_1$ and $P = -bRt+ c_1 = -bk_1t + c_1$ by separation of variables. Here $k_1, c_1$ are integration constants.

Set

$$\frac{dR}{dt} = -bR.$$

$$\frac{dP}{dt} = 0$$

The solution is $R = c_2e^{-bt}$ and $P = k_2$.