Use d'Alembert's Solution to Solve the Wave Equation

[mario99]

$$\frac{\partial^2 y}{\partial t^2} = 81\frac{\partial^2 y}{\partial x^2}$$
$$-\infty < x < \infty, \ \ \ t > 0$$
$$y(x,0) = x^2$$
$$\frac{\partial y}{\partial t}(x,0) = 3$$

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:​

$$y(x,t) = \frac{1}{2}[f(x + ct) + f(x - ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s) \ ds$$

How to use this solution to solve the differential equation directly?
​

 

[MaxWong]

I think you can simply plug in $c = \sqrt{81} = 9$, $f(x) = x^2$, $g(x) = 3$.

 

[mario99]

This means that the solution is:

$\displaystyle y(x,t) = \frac{1}{2}\left[f(x + 9t) + f(x - 9t)\right] + \frac{1}{2*9}\int_{x-9t}^{x+9t} 3 \ ds$


$\displaystyle = \frac{1}{2}\left[(x + 9t)^2 + (x - 9t)^2\right] + \frac{3}{18}\left[(x+9t) - (x-9t)\right]$


$\displaystyle = \frac{1}{2}\left[x^2 + 18tx + 81t^2 + x^2 - 18tx + 81t^2\right] + \frac{1}{6}\left[x+9t - x + 9t\right]$


$\displaystyle = \frac{1}{2}\left[2x^2 + 162t^2\right] + \frac{1}{6}\left[18t\right]$


$\displaystyle = x^2 + 81t^2+ 3t$


Thank you MaxWong. Your help was so Epic.