Question

The solution of \(\frac{\partial^2 u}{\partial t^2} = 4\frac{\partial^2 u}{\partial x^2}\), \(t > 0\), \(-\infty < x < \infty\) satisfying the conditions \(u(x,0) = x\) and \(\frac{\partial u}{\partial t}(x,0) = 0\) is

• A. \(x\)
• B. \(\frac{x^2}{2} + t\)
• C. \(x + 2t\)
• D. \(\frac{t^2}{2} + x\)

Hint:

When initial velocity \(\frac{\partial u}{\partial t}(x,0) = 0\), the solution is simply the average of the initial wave profile shifted left and right.
If \(f(x)\) is linear, the time dependency \(t\) cancels out completely.

Answer 1

The Correct Option is A