Coupled systems and second order ODEs

Exchange between compartments couples their rates. Matrix methods separate total amount from differences between compartments. Second-order equations introduce another type of coupling, between position and velocity, and reveal oscillatory or damped behavior.

1 Constant-coefficient systems

For $x'=Ax$ with a constant real $d\times d$ matrix, $d\ge1$, a real eigenpair $(\lambda,v)$ gives a solution $x(t)=e^{\lambda t}v$. Differentiation verifies it directly. If $A=PDP^{-1}$ has a real eigenbasis, set $x=Pz$: then $z'=Dz$ and

$$x(t)=P\operatorname{diag}(e^{\lambda_1t},\ldots,e^{\lambda_dt})P^{-1}x(0).$$

For arbitrary square $A$, define the matrix exponential

$$e^{tA}=\sum_{k=0}^\infty\frac{t^kA^k}{k!}.$$

A submultiplicative matrix norm bounds the series by $e^{|t|\|A\|}$, giving absolute uniform convergence on bounded time intervals. The derivative series has the same type of bound, so termwise differentiation is justified and $(e^{tA})'=Ae^{tA}$. Multiplying the two series, or solving the derivative identity, gives $e^{tA}e^{-tA}=I$. Thus $x=e^{tA}x_0$ is the unique solution: for any solution, differentiating $e^{-tA}x(t)$ gives zero.

This construction does not assume diagonalizability. For $A=\begin{pmatrix}-1&1\\0&-1\end{pmatrix}=-I+N$ with $N^2=0$, $e^{tA}=e^{-t}(I+tN)$. The factor $t$ cannot be obtained by incorrectly diagonalizing the repeated eigenvalue.

2 Two communicating reservoirs

Two equal-volume, perfectly mixed reservoirs exchange at effective rate $k>0$ and each removes solute at rate $\mu\ge0$. With no external input, concentrations satisfy

$$\binom{c_1'}{c_2'}=\begin{pmatrix}-(k+\mu)&k\\k&-(k+\mu)\end{pmatrix}\binom{c_1}{c_2}.$$

Let $s=c_1+c_2$ and $d=c_1-c_2$. Adding and subtracting gives $s'=-\mu s$ and $d'=-(2k+\mu)d$. Hence

$$c_1(t)=\tfrac12[s_0e^{-\mu t}+d_0e^{-(2k+\mu)t}],\quad c_2(t)=\tfrac12[s_0e^{-\mu t}-d_0e^{-(2k+\mu)t}].$$

When $\mu=0$, total mass is conserved because equal volumes make it proportional to $s$; the difference decays to zero, leaving equal concentrations $s_0/2$. When $\mu>0$, both concentrations eventually vanish, but the imbalance decays faster than the total. For nonnegative initial concentrations and forward time $t\ge0$, $|d_0|\le s_0$ and the faster decay ensures both remain nonnegative.

With unequal constant positive volumes, unweighted concentration sum is not total mass. The total mass is $M=V_1c_1+V_2c_2$. With no external source and the same removal rate $\mu$ in both compartments, exchange cancels and $M'=-\mu M$; it is conserved only when $\mu=0$. Rate coefficients must be derived from the appropriate mass balance.

3 Second-order equations

A linear second-order homogeneous equation is $y''+ay'+by=0$. Substitute $e^{rt}$ to obtain the characteristic equation $r^2+ar+b=0$. Distinct real roots give $C_1e^{r_1t}+C_2e^{r_2t}$; a repeated real root gives $(C_1+C_2t)e^{rt}$; roots $\alpha\pm i\beta$ give $e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t)$. Substitution verifies these solutions, and their independent initial position/velocity values plus uniqueness show they span all solutions.

For the damped oscillator

$$y''+2\gamma y'+\omega_0^2y=0,\qquad\gamma\ge0,\ \omega_0>0,$$

roots are $-\gamma\pm\sqrt{\gamma^2-\omega_0^2}$. The underdamped case $\gamma<\omega_0$ oscillates with angular frequency $\sqrt{\omega_0^2-\gamma^2}$ and envelope $e^{-\gamma t}$. Critical damping has a repeated root; overdamping has two distinct negative real roots.

The energy $E=(y'^2+\omega_0^2y^2)/2$ satisfies

$$E'=y'(y''+\omega_0^2y)=-2\gamma y'^2\le0.$$

This is a direct diagnostic of the sign convention: positive damping must not increase this model energy. The equation is a simplified local model, not a universal description of environmental motion.

Complete real matrix-exponential construction

Fix a real $d\times d$ matrix $A$, $d\ge1$, and use the Euclidean operator norm; its definition gives $\|AB\|\le\|A\|\|B\|$. For $T>0$, on $[-T,T]$ the series terms have norms at most $T^k\|A\|^k/k!$, and their derivatives, for $k\ge1$, have norms at most $\|A\|^kT^{k-1}/(k-1)!$. Each nonnegative majorant converges: beyond a finite index its consecutive-term ratio is at most $1/2$, so its tail is bounded by a geometric series. The same bounds therefore make both matrix series uniformly Cauchy on the interval. Completeness and continuity follow in finitely many real entries, giving continuous limits $E$ and $G$.

For the polynomial partial sums $E_N$, $E_N(t)-I=\int_0^tE_N'(s)ds$. Passing to the limit is justified entrywise by uniform convergence, since the integral error is at most $T$ times a uniform derivative error. Thus $E(t)-I=\int_0^tG(s)ds$. The fundamental theorem now proves differentiability, not merely convergence of a formal derivative series. Reindexing the convergent derivative series gives $G=AE$. Every partial sum commutes with $A$, so $E$ also commutes with $A$.

The product derivative of $E(t)E(-t)$ is $AE(t)E(-t)-E(t)AE(-t)=0$; its value at zero is $I$. Replacing $t$ by $-t$ proves both inverse identities. Hence any solution of $x'=Ax$ has $(E(-t)x(t))'=0$, proving $x(t)=E(t)x(0)$ and uniqueness. For fixed $s$, $E(t+s)$ and $E(t)E(s)$ solve the same equation with the same value at $t=0$, so $E(t+s)=E(t)E(s)$. This also justifies the exponential factorization in variation of constants.

For second-order real equations the three displayed solution pairs have Wronskians $(r_2-r_1)e^{(r_1+r_2)t}$, $e^{2rt}$, and $\beta e^{2\alpha t}$, with $\beta>0$ in the nonreal-root case. They are nonzero. Each initial pair therefore fixes one linear combination. The difference between that combination and an arbitrary solution gives a first-order real system with zero initial value, so the proved uniqueness establishes completeness. The repeated-root pair also follows directly by substituting $y=e^{rt}u$, which reduces the equation to $u''=0$.

In the two-reservoir physical model, positivity is a forward-time conclusion: for $t\ge0$, $e^{-(2k+\mu)t}\le e^{-\mu t}$ and $|d_0|\le s_0$, so each concentration formula is nonnegative. The linear ODE solution itself exists for negative times too, but positivity need not persist backwards. For $k=1$, $\mu=0$, $c_1(0)=1$, $c_2(0)=0$, one has $c_2(t)=(1-e^{-2t})/2<0$ for $t<0$.

4 Graded practice

E1 Foundation. Solve $x'=\operatorname{diag}(-2,1)x$ with $x(0)=(3,4)$ and describe long-time behavior.

E2 Reservoirs. Set $k=1$, $\mu=0$, $c_1(0)=6$, $c_2(0)=2$. Find both concentrations and their limit.

E3 Repeated root. Solve $y''+4y'+4y=0$ with $y(0)=1$, $y'(0)=0$.

E4 Oscillation. Solve $y''+2y'+5y=0$ with $y(0)=0$, $y'(0)=2$.

5 Full solutions

S1. $x(t)=(3e^{-2t},4e^t)$. One component decays while the other grows. A negative eigenvalue in one direction does not make the entire system asymptotically stable.

S2. $s_0=8$, $d_0=4$. Thus $c_1=4+2e^{-2t}$ and $c_2=4-2e^{-2t}$, both tending to 4. Their sum is constantly 8.

S3. The repeated root is $-2$, so $y=(C_1+C_2t)e^{-2t}$. Initial position gives $C_1=1$ and initial derivative gives $C_2-2C_1=0$, so $y=(1+2t)e^{-2t}$.

S4. Roots are $-1\pm2i$. Write $y=e^{-t}(C_1\cos2t+C_2\sin2t)$. Initial position gives $C_1=0$ and derivative gives $2C_2=2$, so $y=e^{-t}\sin2t$.

Summary

Eigenmodes separate coupled linear dynamics, while matrix exponentials handle repeated or defective eigenvalues. Conservation laws identify the right aggregate variables. Second-order equations require two initial conditions, and energy differentiation checks whether damping and signs match the intended physical model.