these operators are commutative. so its easy...
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You mean f(x) = A exp(kx) + B exp(-kx)
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it's a common shorthand to just write the two linearly independent solutions as shown
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You can also do this for a 4th-order DE (useful for describing bending waves in plates):pic.twitter.com/o3i88b8uem
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nice! didn't know that
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The same for a PDE (as you definitely know) like the hyperbolic 1D-wave eqn: f_(xx)-c^(-2)f_(tt)=0 with solution f(x,t)=g(x+/-ct), in general: f(x,t)=g(x+ct)+h(x-ct) with "left and right movers" where f_(x) denotes the first partial derivative of f(x,t) with respect to x etc.
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There are several ways to solve these kind of PDEs: 1) Separation of variables 2) Fourier transform 3) Laplace transform (Also: Method of charactestics after reducing this 2nd order eqn to two 1st order eqns etc).
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This approach can be made perfectly rigorous.
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Lanczos differential operators book deals with this? I bought 4 years ago, and is collecting dust
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