Counterexample bank

False — y′ = $y^{1/3}$, y(0) = 0

False — y′ = $y^{1/3}$, y(0) = 0

False — $f(x) = x^{2}, g(x) = x$|x| on $\mathbb{R}$

False — $y'' + \pi^{2}y = f$ on [0,1] with y(0) = y(1) = 0

False — ẋ $= -y - x^{3}$, ẏ $= x - y^{3}$ at the origin

False — $u u_{x} + u_y = 0$ with u(x, 0) = −x

False — The Tricomi equation $y u_{xx} + u_yy = 0$

False — $f(x) = x^{2}$ at the root 0

False — Runge's function $1/(1 + 25x^{2})$ on [−1, 1]

False — $J[y] = \int_{0}^\pi (1 - (y')^{2})y^{2}$ dx at y ≡ 0

False — $y(x) - 3\int_{0}^{1}$ tx y(t)dt = eˣ

False — A bead on a wire rotating at forced angular velocity $\omega$