separable solution

Solutions may be found by separating variables and seeking a separable solution of the form y(x,t) = X(x)T(t). This leads to the equation and conditions

As X″/X is a function of x alone and T″/T is a function of t alone, both must be a constant k. To meet the boundary conditions X(0) = X(l) = 0, it follows that k = –n2π2/l2 for some positive integer n. The separable solutions are then the normal modes

The above argument applies more generally to other PDEs, and the separable solutions then play an important part in finding the general solution (see superposition priniciple).

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