Use d'Alembert's solution of the wave equation to find the solution of
[MATH]\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0) [/MATH]
[MATH] \frac{\partial u(0,t)}{\partial t} = \alpha \frac{\partial u(0,t)}{\partial x} \qquad (t \geq 0) [/MATH]
[MATH] u(x,0) = \phi_0 (x) \qquad (x \geq 0) [/MATH]
[MATH] \frac{\partial u}{\partial t} (x, 0) = \phi_1 (x) \qquad (x \geq 0) [/MATH]
where [MATH] \alpha \neq -1 [/MATH] is a constant and [MATH] \phi_0 (x) [/MATH] and [MATH] \phi_1 (x) [/MATH] are twice continuously differentiable for [MATH] x > 0 [/MATH] and vanish near [MATH] x = 0 [/MATH]. Show that in general no solution exists when [MATH] \alpha = -1 [/MATH].
[MATH]\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0) [/MATH]
[MATH] \frac{\partial u(0,t)}{\partial t} = \alpha \frac{\partial u(0,t)}{\partial x} \qquad (t \geq 0) [/MATH]
[MATH] u(x,0) = \phi_0 (x) \qquad (x \geq 0) [/MATH]
[MATH] \frac{\partial u}{\partial t} (x, 0) = \phi_1 (x) \qquad (x \geq 0) [/MATH]
where [MATH] \alpha \neq -1 [/MATH] is a constant and [MATH] \phi_0 (x) [/MATH] and [MATH] \phi_1 (x) [/MATH] are twice continuously differentiable for [MATH] x > 0 [/MATH] and vanish near [MATH] x = 0 [/MATH]. Show that in general no solution exists when [MATH] \alpha = -1 [/MATH].