Solution. Replacing \(\varepsilon \) by \(-\varepsilon \) in Theorem 14.2.1, \(\omega = 1 - \frac 38\varepsilon a^2\). The frequency decreases with amplitude. The potential \(\frac 12x^2 - \frac 14\varepsilon x^4\) has maxima at \(x = \pm 1/\sqrt \varepsilon \), saddles of the phase portrait, so oscillations exist only for amplitudes below \(1/\sqrt \varepsilon \), and the period tends to infinity as the orbit approaches the homoclinic orbits; no power series in \(\varepsilon a^2\) can describe that.

Solution. \(h = -(x')^3 = r^3\sin ^3\theta \), so \(r' = -\frac \varepsilon {2\pi }\int _0^{2\pi }r^3\sin ^4\theta \,d\theta = -\frac 38\varepsilon r^3\). Then \(r^{-2} = r_0^{-2} + \frac 34\varepsilon t\), so \(r = r_0/\sqrt {1 + \frac 34\varepsilon r_0^2t}\): the oscillation decays like \(t^{-1/2}\), not exponentially, because cubic damping weakens as the amplitude falls.

Solution. The reduced equation \(-y' + y = 0\) gives \(y = Ce^t\). Now the coefficient of \(y'\) is negative, so the layer is at \(t = 1\); the outer solution satisfies the condition at \(0\): \(y_{\text {out}} = e^t\). Near \(t = 1\), with \(\eta = (1 - t)/\varepsilon \), the inner equation is \(y_{\eta \eta } + y_\eta = 0\) to leading order, so \(y_{\text {in}} = B(1 - e^{-\eta })\) with \(y_{\text {in}}(0) = 0\), and matching to \(y_{\text {out}}(1) = e\) gives \(B = e\). The composite approximation is \(y \approx e^t - e^{1 - (1 - t)/\varepsilon }\).

Solution. Explicit: \(y_{k+1} = (1 - 50h)y_k\), decaying if \(|1 - 50h| < 1\), that is \(h < 0.04\). With \(h = 0.1\), \(y_k = (-4)^k\), a growing oscillation, although the true solution \(e^{-50t}\) decays. Implicit: \(y_{k+1} = y_k/(1 + 50h)\), which decays for all \(h > 0\); with \(h = 0.1\), \(y_k = 6^{-k}\), decaying, though not at the true rate.

Solution. \(x_0' = -x_0\), \(x_0 = e^{-t}\); \(x_1' = -x_1 + x_0^2 = -x_1 + e^{-2t}\), \(x_1(0) = 0\), so \(x_1 = e^{-t} - e^{-2t}\). The exact solution is the Bernoulli solution \(x = 1/\bigl ((1 - \varepsilon )e^t + \varepsilon \bigr )\), which expands to the same first-order terms. Because the unperturbed solution decays to a hyperbolic sink, errors are damped rather than amplified, and the expansion is uniformly valid on \([0, \infty )\). Secular terms arise for oscillators, where the unperturbed system is a centre and the perturbation shifts the frequency.

Solution. \(k_1 = 1\), \(k_2 = 1.05\), \(k_3 = 1.0525\), \(k_4 = 1.10525\), so \(y_1 = 1 + \frac {0.1}{6}(1 + 2.1 + 2.105 + 1.10525) = 1.1051708\overline {3}\). Indeed RK4 applied to \(y' = y\) reproduces the Taylor series \(1 + h + \frac {h^2}2 + \frac {h^3}6 + \frac {h^4}{24} = 1.10517083\), and \(e^{0.1} = 1.10517092\): the local error is \(8.5 \times 10^{-8}\), of order \(h^5/120\).