# Systems of Differential Equations

Physics I · Differential Equations and Vector Calculus · https://tryals.app/learn/physics-i/systems-of-differential-equations

## Coupled Systems & Solutions

Many physical systems involve **coupled** variables where each rate depends on the other. Solving them one at a time destroys the interaction the system models.

A system of two first-order ODEs is written as $\frac{d\vec{x}}{dt} = A\vec{x}$, where $\vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$.

**Solution method**: Find eigenvalues $\lambda_i$ and eigenvectors $\vec{v}_i$ of matrix $A$ to form the general solution:

$$\vec{x}(t) = C_1\,e^{\lambda_1 t}\,\vec{v}_1 + C_2\,e^{\lambda_2 t}\,\vec{v}_2$$

**Key insight**: The eigenvalues of $A$ completely determine long-term behaviour, stability, oscillation, and decay rates.

## Phase Portraits & Example

The eigenvalues classify the **phase portrait** behaviour:

| Eigenvalues | Type | Stability |
| --- | --- | --- |
| $\lambda_1, \lambda_2 < 0$ real | Node | Stable |
| $\lambda_1, \lambda_2 > 0$ real | Node | Unstable |
| $\lambda_1 < 0 < \lambda_2$ real | Saddle | Unstable |
| $\alpha \pm \beta i$, $\alpha < 0$ | Spiral | Stable |
| $\alpha \pm \beta i$, $\alpha > 0$ | Spiral | Unstable |
| $\pm \beta i$ (pure) | Centre | Neutrally stable |

**Worked Example**: A coupled spring-mass system yields a $4 \times 4$ system. Its eigenvalues give the **normal mode frequencies**.

```python
A = np.array([[-3, 1], [2, -2]])
vals, vecs = np.linalg.eig(A)
```

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order the steps for solving a $2\times2$ linear system $X' = AX$.

**Answer:**

1. Write the system in matrix form X' = AX
2. Find eigenvalues of A: $det(A - \lambda I) = 0$
3. Find eigenvectors for each eigenvalue
4. Write general solution $X = c_{1}v_{1}e^{\lambda_{1}t} + c_{2}v_{2}e^{\lambda_{2}t}$
5. Apply initial conditions to find $c_{1}$ and $c_{2}$

**Why:** Eigenvalues and eigenvectors of A determine the system's solution.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/order-the-steps-for-solving-a-2-2-linear-system-x-ax

### 2. Rabbits $R$ and foxes $F$: $R' = aR - bRF$, $\;F' = -cF + dRF$. What does the $RF$ term represent?

A. Competition among rabbits for food
B. Encounters between the two species
C. Random environmental noise
D. The combined biomass of the ecosystem

**Answer:** B. Encounters between the two species

**Why:** The product $RF$ counts meetings: doubling either population doubles encounters. It drains rabbits ($-bRF$) and feeds foxes ($+dRF$), coupling the equations into the predator–prey cycles Lotka and Volterra made famous.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/rabbits-r-and-foxes-f-r-ar-brf-f-cf-drf-what-does-the

### 3. Predator–prey populations cycle. Starting from **many rabbits, few foxes**, order the phases of one full cycle.

**Answer:**

1. Abundant prey lets the fox population grow
2. Many foxes drive the rabbit population down
3. Prey scarcity starves the fox population down
4. With few foxes, the rabbit population recovers

**Why:** Each species chases the other’s state a quarter-cycle behind, producing closed loops in the phase plane rather than settling. Real lynx–hare fur records from the Hudson Bay Company show precisely these lagged oscillations.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/predator-prey-populations-cycle-starting-from-many-rabbits-few

### 4. If both eigenvalues of a $2\times2$ system have negative real parts, the equilibrium is asymptotically stable.

**Answer:** True

**Why:** Negative real parts $\to$ all solutions decay $\to$ asymptotically stable.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/if-both-eigenvalues-of-a-2-2-system-have-negative-real-parts-the

### 5. A system of coupled ODEs can be decoupled by changing to the eigenvector basis. Why does that make it solvable?

A. Because the transformation linearises an otherwise nonlinear system
B. Because eigenvectors are orthogonal and orthogonality removes the coupling
C. Because in that basis each equation involves one variable
D. Because the eigenvector basis has fewer variables than the original

**Answer:** C. Because in that basis each equation involves one variable

**Why:** Diagonalising decouples. The normal modes of a coupled oscillator are exactly these eigen-directions, combinations that move independently while the raw coordinates do not.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/a-system-of-coupled-odes-can-be-decoupled-by-changing-to-the

### 6. Which equilibrium types are stable for a 2D linear system?

A. Stable node (both eigenvalues negative real)
B. Saddle point
C. Center
D. Stable spiral (complex eigenvalues with negative real part)

**Answer:** A. Stable node (both eigenvalues negative real); D. Stable spiral (complex eigenvalues with negative real part)

**Why:** Centers are neutrally stable but not asymptotically stable; saddles are unstable.

Page: https://tryals.app/practice/physics-i/systems-of-differential-equations/which-equilibrium-types-are-stable-for-a-2d-linear-system