Solution. \(b^2 - ac = -x\): elliptic for \(x > 0\), parabolic on \(x = 0\), hyperbolic for \(x < 0\). The slopes satisfy \(m^2 + x = 0\), so \(dy/dx = \pm \sqrt {-x}\) and the characteristics are \(y = y_0 \mp \frac 23(-x)^{3/2}\), the Tricomi curves with the roles of \(x\) and \(y\) exchanged.

Solution. \(b = -1\), \(a = c = 1\), so \(b^2 - ac = 0\). The slope equation \(m^2 + 2m + 1 = 0\) has the double root \(m = -1\), so \(\xi = x + y\) is characteristic (check: \(1 - 2 + 1 = 0\)). Take \(\eta = x\). Then \(A = B = 0\) and \(C = 1\), so the equation is \(u_{\eta \eta } = 0\) and \[ u = F(x + y) + xG(x + y) . \] Check for the second term: \(u_{xx} = 2G' + xG''\), \(u_{xy} = G' + xG''\), \(u_{yy} = xG''\), and \(2G' + xG'' - 2G' - 2xG'' + xG'' = 0\).

Solution. \(b^2 - ac = 1 - 5 = -4 < 0\). The slopes are \(m = 1 \pm 2i\), so \(\varphi = y - (1 + 2i)x\) is a complex characteristic, with \(\alpha = y - x\) and \(\beta = 2x\) (the sign of \(\beta \) is immaterial). Then \(A = 1 - 2 + 5 = 4\), \(C = 4\), and \(B = 1\cdot (-1)(2) + 1\cdot \bigl ((-1)\cdot 0 + 1\cdot 2\bigr ) + 5\cdot 1\cdot 0 = 0\), so the equation is \(4(u_{\alpha \alpha } + u_{\beta \beta }) = 0\).

Solution. The coefficient of \(u_{xx}\) is \(-c^2\), that of \(u_{tt}\) is \(1\), and there is no mixed term, so the discriminant is \(c^2 > 0\): hyperbolic. The characteristics satisfy \((dx/dt)^2 = c^2\), so they are the lines \(x \pm ct = \text {const}\). With \(\xi = x + ct\), \(\eta = x - ct\), \(u_{tt} - c^2u_{xx} = -4c^2u_{\xi \eta }\), so \(u_{\xi \eta } = 0\) and \(u = F(x + ct) + G(x - ct)\): a wave moving left and a wave moving right at speed \(c\). This is the starting point of the next section on the wave equation.

Solution. The solution is \(u_n = e^{-\sqrt n}\sin (nx)\sinh (ny)/n\). The \(k\)th \(x\)-derivative of the data is bounded by \(n^ke^{-\sqrt n} \to 0\). But \(\sup _x|u_n(x, y)| = e^{-\sqrt n}\sinh (ny)/n \geq e^{ny - \sqrt n}/(4n)\) for \(ny \geq 1\), and \(ny - \sqrt n - \log n \to \infty \).