Задача 78916 Надо решить задачу , номер 5...
Условие
Решение
[m]\large \frac{ ∂ u}{ ∂ x} = e^{-\cos(x - at)} \cdot \sin(x - at)[/m]
[m] \frac{ ∂^2 u}{ ∂ x^2} = e^{-\cos(x - at)} \cdot \sin(x - at) \cdot \sin(x - at) + e^{-\cos(x - at)} \cdot \cos(x - at) [/m]
[m]\large \frac{ ∂^2 u}{ ∂ x^2} = e^{-\cos(x - at)} \cdot (\sin^2 (x - at) + \cos(x - at))[/m]
[m]\large \frac{ ∂ u}{ ∂ t} = e^{-\cos(x - at)} \cdot \sin(x - at) \cdot (-a) = -ae^{-\cos(x - at)} \cdot \sin(x - at)[/m]
[m] \frac{ ∂^2 u}{ ∂ t^2} = -ae^{-\cos(x - at)} \cdot \sin(x - at) \cdot \sin(x - at) \cdot (-a) -[/m]
[m]- ae^{-\cos(x - at)} \cdot \cos(x - at) \cdot (-a) [/m]
[m]\large \frac{ ∂^2 u}{ ∂ t^2}= a^2e^{-\cos(x - at)} \cdot (\sin^2 (x - at) + \cos(x - at))[/m]
Получаем:
[m]\large a^2 \cdot \frac{ ∂^2 u}{ ∂ x^2} = \frac{ ∂^2 u}{ ∂ t^2}[/m]
Доказано.