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The properties of certain linear and nonlinear differential equations. (English) Zbl 1450.34064
Singh, Vinai K. (ed.) et al., Advances in mathematical methods and high performance computing. Cham: Springer. Adv. Mech. Math. 41, 193-200 (2019).
The paper deals with linear differential operators of the third and second orders. In those cases when this operators have a common zero differential relations arise between the coefficients of the operators. For example, (Theorem 3) if \(L=\delta^3+p(z)\delta^2+q(z)\delta+r(z), L_1=\delta^2+q_1(z)\delta+r_1(z)\) and relations \[ r_1' =r-pr_1+q_1r_1, \quad q_1' =q-pq_1+q_1^2-r_1 \] hold then \(\mathrm{Ker } L \supset\mathrm{Ker }L_1\). To obtain these relations, the authors use the Shwarz derivative of the ratio of two linearly independent zeros of the operator (see [N. A. Lukashevich, Differ. Equations 35, No. 10, 1384–1390 (1999; Zbl 1020.34004); translation from Differ. Uravn. 35, No. 10, 1366–1371 (1999)]). Unfortunately, Theorem 1 as stated in this paper is not true. Since it immediately follows that any two third order operators having a common zero coinside. For example, let \(\delta=\frac{d}{dz}, L=(\delta^2+1)(\delta -1),L_1=(\delta^2+g(z))(\delta -1), g(z)\ne \mathrm{const}, \xi=\frac{\sin(z)}{\exp(z)}, w=S\xi.\) Then equations \(L=0\) and \(L_1=0\) satisfy the conditions of Theorem 1, but \(L\ne L_1\).
For the entire collection see [Zbl 06982489].
MSC:
34M03 Linear ordinary differential equations and systems in the complex domain
12H05 Differential algebra
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References:
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