{
  "id": "2006.11592",
  "version": "v1",
  "published": "2020-06-20T15:09:02.000Z",
  "updated": "2020-06-20T15:09:02.000Z",
  "title": "Viewing nonoscillatory second order linear differential equations from the angle of Riccati equations",
  "authors": [
    "Jaroslav Jaros",
    "Takashi Kusano",
    "Tomoyuki Tanigawa"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We build an existence theory for nonoscillatory second order differential equations of the form (A) $(p(t)x')' = q(t)x, $ $p(t)$ and $q(t)$ being positive continuous functions on $[a,\\infty)$, in which a crucial role is played by a pair of the Riccati differential equations (R1) $u' = q(t) - u^2/p(t)$, (R2) $ v' = 1/p(t) - q(t)v^2$, associated with (A). An essential part of the theory is the construction of a pair of linearly independent nonoscillatory solutions $x_1(t)$ and $x_2(t)$ of (A) enjoying explicit exponential-integral representations in terms of solutions $u_1(t)$ and $u_2(t)$ of (R1) or in terms of solutions $v_1(t)$ and $v_2(t)$ of (R2).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-20T15:09:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C10"
    ],
    "keywords": [
      "second order linear differential equations",
      "nonoscillatory second order linear differential",
      "viewing nonoscillatory second order linear",
      "second order differential equations",
      "riccati equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}