[math]\frac{\partial^2 y}{\partial t^2} = 81\frac{\partial^2 y}{\partial x^2}[/math]
[math]-\infty < x < \infty, \ \ \ t > 0[/math]
[math]y(x,0) = x^2[/math]
[math]\frac{\partial y}{\partial t}(x,0) = 3[/math]
[math]y(x,t) = \frac{1}{2}[f(x + ct) + f(x - ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s) \ ds[/math]
[math]-\infty < x < \infty, \ \ \ t > 0[/math]
[math]y(x,0) = x^2[/math]
[math]\frac{\partial y}{\partial t}(x,0) = 3[/math]
I know how to solve this problem from scratch, but I don't know how to solve it by d'Alembert's solution.
The d'Alembert's solution is:
The d'Alembert's solution is:
[math]y(x,t) = \frac{1}{2}[f(x + ct) + f(x - ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s) \ ds[/math]
How to use this solution to solve the differential equation directly?