Common Pitfalls

:::caution Common Pitfall When using undetermined coefficients, always check whether your guess Overlaps with the homogeneous solution. For $y"' - 4y = e^{2x}$ Guessing $y_p = Ae^{2x}$ fails Because $e^{2x}$ satisfies the homogeneous equation. You must use $y_p = Axe^{2x}$ instead. :::

:::caution Common Pitfall The Laplace transform of $y'(t)$ is $sY(s) - y(0)$ Not $sY(s)$ . The Initial conditions are built into the transform. Forgetting them leads to incorrect solutions. :::

:::caution Common Pitfall Separation of variables can miss solutions. When dividing by $h(y)$ to Separate, check whether $h(y) = 0$ gives any valid solutions. For example, solving $y' = y^2$ by Separating gives $y = -1/(x + C)$ But misses the solution $y = 0$ . :::

:::caution Common Pitfall The Fourier series of a function converges to the function only at points Of continuity. At jump discontinuities, it converges to the midpoint of the jump. The Gibbs Phenomenon causes overshoots near jumps that do not vanish as more terms are added. :::

:::caution Common Pitfall When solving PDEs by separation of variables, the boundary conditions Determine the eigenvalues. Using the wrong boundary conditions (e.g., Neumann instead of Dirichlet) Leads to a completely different set of eigenfunctions and eigenvalues. :::

:::caution Common Pitfall Not every first-order ODE can be solved analytically. Equations like $y' = x^2 + y^2$ have no closed-form solution in terms of elementary functions. Numerical methods (Euler, Runge-Kutta) may be necessary. :::

:::caution Common Pitfall The linearization of a nonlinear system near a critical point is only Valid for hyperbolic critical points (no eigenvalues on the imaginary axis). If eigenvalues lie on The imaginary axis, the nonlinear system can behave very differently from its linearization. :::

:::caution Common Pitfall When computing the inverse Laplace transform, always check that the Partial fraction decomposition is correct before inverting term-by-term. A common error is Forgetting to include all powers of irreducible quadratic factors. :::

:::caution Common Pitfall For the Euler-Cauchy equation $x^2 y'' + axy' + by = 0$ Remember that $x^r$ with complex $r = \alpha \pm i\beta$ gives solutions involving $\cos(\beta \ln x)$ and $\sin(\beta \ln x)$ Not $\cos(\beta x)$ and $\sin(\beta x)$ . :::

:::caution Common Pitfall The Wronskian $W(y_1, y_2)$ being zero at a single point does not Necessarily mean the solutions are linearly dependent. For linear ODEs with continuous coefficients, $W \equiv 0$ everywhere or $W \neq 0$ everywhere. Check Abel’s identity. :::

9.9 Worked Example: Lyapunov Function

Problem. Show that $(0, 0)$ is asymptotically stable for the system $x' = -x - y^3$ , $y' = x^3 - y$ .

Solution

Solution. Critical point: $-x - y^3 = 0$ and $x^3 - y = 0$ gives $x = 0, y = 0$ .

Choose the Lyapunov function $V(x, y) = \frac{x^2}{2} + \frac{y^2}{2}$ . $V(0,0) = 0$ and $V > 0$ for $(x, y) \neq (0, 0)$ .

$\dot{V} = \frac{\partial V}{\partial x}x' + \frac{\partial V}{\partial y}y' = x(-x - y^3) + y(x^3 - y)$

$= -x^2 - xy^3 + x^3 y - y^2 = -x^2 - y^2 + xy(x^2 - y^2)$

This is not negative definite. Let us try $V(x, y) = \frac{x^4}{4} + \frac{y^4}{4}$ :

$\dot{V} = x^3(-x - y^3) + y^3(x^3 - y) = -x^4 - x^3 y^3 + x^3 y^3 - y^4 = -(x^4 + y^4)$ .

Since $\dot{V} = -(x^4 + y^4) \lt 0$ for $(x, y) \neq (0, 0)$ The origin is asymptotically Stable by Lyapunov’s theorem. $\blacksquare$

9.10 Worked Example: Van der Pol Oscillator

Problem. Show that the Van der Pol equation $x'' + \mu(x^2 - 1)x' + x = 0$ with $\mu > 0$ Has a unique limit cycle.

Solution

_Solution.* Write as a system: $x' = y$ , $y' = -x - \mu(x^2 - 1)y$ .

The only critical point is $(0, 0)$ . The Jacobian at the origin is $J = \begin{pmatrix} 0 & 1 \\ -1 & \mu \end{pmatrix}$ .

$\mathrm{tr}(J) = \mu > 0$ , $\det(J) = 1 > 0$ , $\tau^2 - 4\Delta = \mu^2 - 4$ .

For $0 \lt \mu \lt 2$ : complex eigenvalues with positive real part (unstable spiral). For $\mu \geq 2$ : two positive real eigenvalues (unstable node).

The origin is always unstable. To show existence of a limit cycle, we use a Liénard’s theorem Or construct a trapping region. Define

$L(x) = x^3/3 - x$ and write the system as $x' = y$ , $y' = -x - \mu L'(x) y$ .

The function $L(x)$ has zeros at $x = \pm 1$ . For $\mu > 0$ The damping is negative for $\lvert x \rvert \lt 1$ (energy input) and positive for $\lvert x \rvert > 1$ (energy Dissipation). This creates a unique stable limit cycle that passes through $x = \pm 1$ .

A detailed …/1-number-and-algebra/3_proof-and-logic requires constructing an annular region and applying the Poincaré-Bendixson Theorem (the inner boundary encloses the unstable origin; the outer boundary is chosen so that Trajectories point inward). $\blacksquare$

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