Abstract We consider the Cauchy problem ∂ψ/∂t = - ⊺ψ, ψ(x, 0) = f(x), where ⊺ is a ordinary differential operator in x (of order at least 2), x belongs to an unbounded interval ∣ ⊂ R, and fϵL1(∣). The fact that f does not belong to L2(∣). together with the general nature of the differential expression ⊺ preclude the application of classical methods in the case where the order of ⊺ is more than 2; instead we use a continuous spectrum eigenfunction expansion, developed in the paper, to obtain a solution ψf of the problem which depends continuously on f in an appropriate sense and also converges to f in a natural topology as t → 0. The solution depends upon a kernel function which may, in particular cases, be calculated explicitly. Questions of approximation of the solution by finite sums are also considered.