# solving ODE equations (time evolutions of probability)

I have 2 ode equations

$\frac{dP_1}{dt}=R_{12}P_2-R_{21}P_1$

$\frac{dP_2}{dt}=R_{21}P_1-R_{12}P_2$

$R_{12}=2R_{21}$ initial conditions are $P_1=0.1, P_2=0.9$

I solved this with numerical solution by using matlab (ODE45 function)

So I know P1 =0.666 and P2 = 0.333

And I tried to solve this with Laplace transform

so the

$P_2^{laplace} = \frac{P_2(0)s+R_{21}(P_2(0)+P_1(0))}{s^2+sR_{12}+sR_{21}}$

however, is there are any other way to solve this ?

The prof. mentioned about the diagonal matrix, but I dont know how to approach...

Turn it into a matrix differential equation (note that I've substituted $A=R_{21}$ and $B=R_{12}$): $$\frac{d}{dt} \left[ \begin{array}{cc} P_1\\ P_2 \end{array}\right] = \left[\begin{array}{c} -A & B\\ A & -B \end{array}\right]\left[ \begin{array}{c} P_1\\ P_2 \end{array}\right].$$ Then solve for eigenvalues: $0=(A+\lambda)(B+\lambda)-AB$. So $\lambda=0$ and $-(A+B)$. Find the eigenvectors $\overset{\rightarrow}{\xi}_1$ and $\overset{\rightarrow}{\xi}_2$: $$\overset{\rightarrow}{\xi}_1=\left[ \begin{array}{c} 1\\ -1 \end{array}\right] \quad \text{ and } \quad \overset{\rightarrow}{\xi}_2=\left[ \begin{array}{c} B\\ A \end{array}\right]$$ Construct the solution: $$\left[ \begin{array}{c} P_1\\ P_2 \end{array}\right]= c_1\overset{\rightarrow}{\xi}_1\exp{\left(-(A+B)t\right)}+c_2\overset{\rightarrow}{\xi}_2.$$ Then you can plug in whatever initial conditions you like and solve for constants $c_1$ and $c_2$. If you like, the solutions can be written separately as $$P_1(t)=c_1e^{-(A+B)t}+c_2B$$ $$P_2(t)=-c_1e^{-(A+B)t}+c_2A$$
However, notice that adding your equations give $P_1'(t)+P_2'(t)=0$, so $P_2(t)=-P_1(t)+C$. This reduces the equations for $P_1$ to $$\frac{d}{dt}P_1=R_{12}(C-P_1)+R_{21}P_1$$ which can be solved using an integrating factor. Then initial values can be plugged in and we can solve for $P_2$ as well.