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A Class of Homogeneous Linear Differential Equations of the Second Order

Published online by Cambridge University Press:  03 November 2016

F. Underwood
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Extract

1. In a recent paper Mammana considered the equation

z″ + pz′ + qz = 0,.....(1)

where x is the independent variable and p and q are functions of x, real, finite and continuous in the interval (a, b), and investigated the possibilities of its decomposition into the form

(D + α)(D + β)z = 0,.....(2)

where D ≡ d/dx and α and β are functions of x.

Type
Research Article
Information
The Mathematical Gazette , Volume 16 , Issue 218 , May 1932 , pp. 122 - 125
Copyright
Copyright © Mathematical Association 1932

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References

page no 122 note * Mammana, G., “Sopra un nuovo metodo di studio delle equazioni differenziali lineari”, Math. Zeit, 25 (1926), 734–748 CrossRefGoogle Scholar. This work is continued and extended to equations of the nth. order in a more recent paper, “Decomposizione delle espressioni differenziali lineari omogenee in prodotti di fattori simbolici e applicazione relativa allo studio delle equazioni differenziali lineari”, Math. Zeit., 33 (1931), 186-231.

page no 123 note * It will be seen, of course, that the first class is merely that special case of the second for which c = 0, but it seemed to be worthy of separate consideration.

page no 124 note * The fourth class reduces to the first when ψ = 0, and to the second when ψ = c.