Abstract
In this paper, we use the Laplace transform technique to examine the generalized solutions of the nth order Cauchy–Euler equations. By interpreting the equations in a distributional way, we found that whether their solution types are classical, weak or distributional solutions relies on the conditions of their coefficients. To illustrate our findings, some examples are exhibited.
Highlights
The nth order Cauchy–Euler equations an tn y(n) (t) + an−1 tn−1 y(n−1) (t) + · · · + a1 ty0 (t) + a0 y(t) = 0, (1)where a0, a1, . . . , an are real constant coefficients and t ∈ R, are often one of the first higher order ordinary homogeneous linear differential equations with variable coefficients introduced in an undergraduate level course
Where a0, a1, . . . , an are real constant coefficients and t ∈ R, are often one of the first higher order ordinary homogeneous linear differential equations with variable coefficients introduced in an undergraduate level course
Replacing y with tr in the Cauchy–Euler equations yields the characteristic polynomial whose roots determine the forms of the general solution
Summary
They can arise as a distributional solution for certain classes of ordinary differential equations with singular coefficients He constructed a formula for m corresponding to each type of generalized solution of Equation (3), which are Laplace transformable. Nanta [17] studied the distributional solutions of the nth order Cauchy–Euler an tn y(n) (t) + an−1 tn−1 y(n−1) (t) + · · · + a1 ty0 (t) + a0 y(t) = 0, (5)
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