Chapter 4 · Systems of DEs

Applications: Two-Tank Systems & Engineering

Where Systems of DEs Meet the Real World

Dr. Mohamed Mabrok · Qatar University

Why Applications Matter

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Chemical Engineering

Mixing tanks, concentration dynamics, and chemical reactions governed by conservation laws.

\frac{dh_i}{dt} = \text{inflow} - \text{outflow}

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Mechanical Engineering

Coupled springs, vibrations, and oscillatory systems with energy transfer.

m_ix_i'' + \text{forces} = 0

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Electrical Engineering

Multi-loop circuits, coupled inductors, and resonance phenomena.

\sum V = 0 \text{ (Kirchhoff)}

The Common Thread

      All governed by: $\mathbf{x}' = A\mathbf{x} + \mathbf{b}(t)$ — systems of first-order linear DEs!

The Two-Tank System

Physical Setup

      Tank 1: Gets external inflow $Q_{\text{in}}$, drains to Tank 2 at rate proportional to height ($k_1 h_1$)
    
      Tank 2: Receives flow from Tank 1, drains out at rate proportional to its height ($k_2 h_2$)
    
      The Coupling: The outflow from Tank 1 is the inflow to Tank 2 — $k_1 h_1$ appears in BOTH equations!

Key Insight

      This is not two independent ODEs. The system is coupled — what happens in Tank 1 directly affects Tank 2.

Governing Equations (Derivation)

Conservation of Mass

      $$A_1 \frac{dh_1}{dt} = Q_{\text{in}} - k_1 h_1$$
    
      $$A_2 \frac{dh_2}{dt} = k_1 h_1 - k_2 h_2$$
    
      where $A_i$ = tank area, $k_i$ = valve coefficients, $h_i$ = height, $Q_{\text{in}}$ = external inflow rate

Matrix Form

      $$\frac{d}{dt}\begin{pmatrix} h_1 \\ h_2 \end{pmatrix} = \begin{pmatrix} -k_1/A_1 & 0 \\ k_1/A_2 & -k_2/A_2 \end{pmatrix} \begin{pmatrix} h_1 \\ h_2 \end{pmatrix} + \begin{pmatrix} Q_{\text{in}}/A_1 \\ 0 \end{pmatrix}$$
    
      Lower-triangular matrix: Tank 1 is independent, Tank 2 depends on Tank 1.

Eigenvalue Analysis: Stability & Steady State

Eigenvalues (Lower Triangular)

      $$\lambda_1 = -\frac{k_1}{A_1}, \quad \lambda_2 = -\frac{k_2}{A_2}$$
    
      Both ALWAYS NEGATIVE because $k_i, A_i > 0$. The system is always STABLE!

Steady State

      $$h_1^* = \frac{Q_{\text{in}}}{k_1}, \quad h_2^* = \frac{Q_{\text{in}}}{k_2}$$
    
      Steady state depends on valve coefficients, NOT tank areas. Areas only affect how fast we reach it.

Worked Example

Example 1: Fill from Empty

Parameters

      $A_1 = A_2 = 2$ m², $k_1 = k_2 = 0.5$ m²/s, $Q_{\text{in}} = 1$ m³/s, $h_1(0) = h_2(0) = 0$

STEP 1 Find eigenvalues

\lambda_1 = \lambda_2 = -0.25 \text{ (repeated!)}

STEP 2 Tank 1 is independent

h_1(t) = 2(1 - e^{-0.25t})

STEP 3 Tank 2 with input from Tank 1

h_2(t) = 2 - 2e^{-0.25t} - 0.5te^{-0.25t}

Physical Insight

      Tank 1 reaches steady state ($h_1^* = 2$ m) first. Tank 2 lags behind, approaching the same steady state as Tank 1 fills it faster than it drains.

Worked Example

Example 2: Drainage

Parameters

      $Q_{\text{in}} = 0$ (no external input), $h_1(0) = 4$ m, $h_2(0) = 3$ m, same tank parameters as before.

STEP 1 Tank 1 simple decay

h_1(t) = 4e^{-0.25t}

STEP 2 Tank 2 — surprising result!

h_2(t) = (3 + t)e^{-0.25t}

The Fascinating Behavior

      Tank 2 initially rises before draining! The term $te^{-0.25t}$ creates a "bump." Initially, inflow from Tank 1 exceeds Tank 2's outflow, so it fills momentarily. Then, as Tank 1 empties faster, Tank 2 starts draining.

Worked Example

Example 3: Separation of Timescales

Parameters

      $k_1 = 1$, $k_2 = 0.25$, $A_1 = A_2 = 2$, $Q_{\text{in}} = 1$. Distinct valve coefficients!

STEP 1 Find eigenvalues

\lambda_1 = -0.5, \quad \lambda_2 = -0.125

STEP 2 Timescale ratio

\frac{\lambda_1}{\lambda_2} = 4 \text{ — Tank 1 responds 4 times faster!}

STEP 3 Steady states

h_1^* = 1 \text{ m}, \quad h_2^* = 4 \text{ m}

Insight

      Small drain valve in Tank 2 allows it to fill much higher. Tank 1 reaches equilibrium quickly, then Tank 2 slowly fills to its steady state.

Coupled Spring-Mass System

Physical Setup

      Two masses connected by three springs: left wall — $k_1$ — mass 1 — $k_2$ — mass 2 — $k_3$ — right wall

Force Balance

      $$m_1 x_1'' = -k_1 x_1 + k_2(x_2 - x_1) = -(k_1+k_2)x_1 + k_2 x_2$$
    
      $$m_2 x_2'' = -k_2(x_2 - x_1) - k_3 x_2 = k_2 x_1 - (k_2+k_3)x_2$$
    
      The middle spring $k_2$ couples the two equations — it creates the coupling term!

Convert to First-Order System

      Let $x_1, v_1 = x_1', x_2, v_2 = x_2'$ (4 variables). The system becomes $\mathbf{x}' = A\mathbf{x}$ with $A \in \mathbb{R}^{4 \times 4}$. Eigenvalues from $A$ determine if motion is oscillatory or decays.

Normal Modes of Vibration

Simplified Case

      $m_1 = m_2 = 1$, $k_1 = k_3 = 2$, $k_2 = 1$. Analyze the 2×2 displacement system.

STEP 1 System matrix

A = \begin{pmatrix} -3 & 1 \\ 1 & -3 \end{pmatrix}

STEP 2 Find eigenvalues

\lambda_1 = -2, \quad \lambda_2 = -4

STEP 3 Eigenvectors (normal modes)

\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \text{ (in-phase)}, \quad \mathbf{v}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \text{ (out-of-phase)}

Physical Interpretation

      In-phase mode: Both masses move together (lower frequency). Out-of-phase mode: Masses move opposite (higher frequency). These are the two independent ways the system can vibrate!

RLC Circuit Systems

Multi-Loop Circuits

      Apply Kirchhoff's voltage law ($\sum V = 0$) to each loop. Shared components (resistors, inductors) couple the equations.

For Two-Loop Circuits

      Let $i_1, i_2$ be the loop currents. Kirchhoff gives:
    
      $$L_1 i_1' + R_1 i_1 + M i_2' = V_1(t)$$
    
      $$L_2 i_2' + R_2 i_2 + M i_1' = V_2(t)$$
    
      The mutual inductance $M$ couples the loops. The system matrix encodes the circuit topology.

Eigenvalues Determine Behavior

      Real eigenvalues → exponential decay (overdamped). Complex eigenvalues → oscillation (underdamped). The real part determines if energy grows or decays.

Population Dynamics: Predator-Prey

Lotka-Volterra Equations

      $$\frac{dx}{dt} = \alpha x - \beta xy$$
    
      $$\frac{dy}{dt} = \delta xy - \gamma y$$
    
      $x$ = prey population, $y$ = predator population. Products $xy$ couple the equations.

Near Equilibrium (Linearization)

      At equilibrium $x^* = \gamma/\delta, y^* = \alpha/\beta$, linearize to get:
    
      $$\frac{d}{dt}\begin{pmatrix} x - x^* \\ y - y^* \end{pmatrix} = \begin{pmatrix} 0 & -\beta x^* \\ \delta y^* & 0 \end{pmatrix} \begin{pmatrix} x - x^* \\ y - y^* \end{pmatrix}$$

Eigenvalues

      Purely imaginary eigenvalues! This means populations cycle — they don't approach equilibrium. This is the famous predator-prey oscillation.

Modeling Strategy: From Physics to Matrix Form

Five-Step Process

      Step 1: Identify state variables. What changes over time? (heights, positions, currents, populations...)
    
      Step 2: Write conservation/balance laws. Apply physical principles (mass balance, force, voltage, etc.) to each variable.
    
      Step 3: Identify couplings. Which variables affect each other? These create off-diagonal entries.
    
      Step 4: Write $\mathbf{x}' = A\mathbf{x} + \mathbf{b}$. Convert to matrix form with constant coefficient matrix $A$.
    
      Step 5: Compute eigenvalues & eigenvectors. Predict long-term behavior: stable? oscillatory? growing?

Complete Worked Example

Two-Tank System with ICs

Problem

      $A_1 = 1$, $A_2 = 3$, $k_1 = 2$, $k_2 = 1$, $Q_{\text{in}} = 0$ (no external input), $h_1(0) = 6$ m, $h_2(0) = 2$ m. Solve the system.

STEP 1 Write the system

A = \begin{pmatrix} -2 & 0 \\ 2/3 & -1/3 \end{pmatrix}

STEP 2 Find eigenvalues

\lambda_1 = -2, \quad \lambda_2 = -1/3

STEP 3-4 Find eigenvectors & build general solution

\mathbf{x}(t) = C_1 e^{-2t} \begin{pmatrix} 1 \\ 1 \end{pmatrix} + C_2 e^{-t/3} \begin{pmatrix} 0 \\ 1 \end{pmatrix}

STEP 5 Apply initial conditions

h_1(t) = 6e^{-2t}, \quad h_2(t) = 6e^{-2t} + (2 - 6)e^{-t/3} = 6e^{-2t} - 4e^{-t/3}

Connecting Math to Physics

Mathematical Features

          Negative eigenvalues: System returns to equilibrium (stable)
        
          Eigenvalue magnitudes: Rate of decay. Larger magnitude = faster decay
        
          Eigenvectors: Independent modes of behavior. Show how state variables move together
        
          Complex eigenvalues: System oscillates while decaying or growing

Engineering Implications

          Valve sizing: Choose $k_i$ to control response speed
        
          Avoid resonance: In mechanical systems, keep driving frequency away from natural frequencies
        
          Stability design: Ensure all eigenvalues have negative real parts (BIBO stable)
        
          Time constants: $\tau = -1/\lambda$ tells you settling time

Common Modeling Mistakes

Watch Out For...

Forgetting units: Tank example uses m³/s and m². Mixing units leads to wildly incorrect coefficients.
Sign errors in coupling: Outflow from Tank 1 is INFLOW to Tank 2. Get the sign right or your equations are backwards.
Ignoring physical reasonableness: If you solve for negative concentration or height, check your work! Physical variables must stay positive.
Confusing steady state with initial condition: $\mathbf{x}^*$ (equilibrium) is where the system goes, not where it starts.
Forgetting second-order → first-order conversion: Spring systems need both position and velocity as state variables, doubling the system size.
Eigenvalues don't tell the full story: You need eigenvectors too! The eigenvalues give rates, eigenvectors give directions.

Key Takeaways

Two-Tank Systems

One-way coupling (lower-triangular). Always stable (negative eigenvalues). Steady state depends on valve sizes, not tank areas.

Spring-Mass & Circuits

Coupling through shared components. Normal modes from eigenvectors. Frequency from eigenvalues. Can oscillate or decay.

Universal Method

Model → Matrix $A$ → Eigenvalues/vectors → Predict behavior. Works for any coupled system!

The Big Picture

      Systems of DEs are everywhere in engineering: tanks, springs, circuits, populations, chemical reactions, and more. The eigenvalue method is your universal tool for understanding how ANY coupled system behaves. Learn this, and you can analyze almost any real-world phenomenon!

Next Up

      Exam Review — bringing everything together across all of Chapter 4. You now know how to model, solve, and interpret systems of DEs in context!