Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Every IVP has a unique solution” — false

Counterexample: y′ = y^{1/3}, y(0) = 0

Not Lipschitz at 0; y ≡ 0 and y = $(2 x /3)^{3/2}$ both solve it.

ODE

#2 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“A continuous right-hand side gives a unique solution” — false

Counterexample: y′ = y^{1/3}, y(0) = 0

Not Lipschitz at 0: y ≡ 0 and y = $(2 x /3)^{3/2}$ both solve it, as do infinitely many hybrids.

ODE

#3 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“A locally unique solution exists for all time” — false

Counterexample locked — unlock with Notes + PYQ

ODE

#4 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“W(f, g) ≡ 0 implies f and g are linearly dependent” — false

Counterexample: f(x) = x², g(x) = x|x| on ℝ

The Wronskian vanishes identically but no constant multiple relates them. The implication holds only for solutions of a common linear ODE.

ODEwronskian

#5 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Every boundary value problem has a Green's function” — false

Counterexample: y″ + π²y = f on [0,1] with y(0) = y(1) = 0

$λ = π^{2}$ is an eigenvalue of the homogeneous problem, so the operator is not invertible and no Green's function exists.

ODEgreen

#6 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Linearisation determines stability at every equilibrium” — false

Counterexample: ẋ = −y − x³, ẏ = x − y³ at the origin

The linearisation is a centre (eigenvalues ±i), but $V = x^{2} + y^{2}$ gives V̇ $= - 2 (x^{4} + y^{4}) < 0$ : asymptotically stable. Non-hyperbolic equilibria escape Hartman–Grobman.

ODEstability