ピコーンの等式

u と v を、二つの自己随伴形式の二階同次線型微分方程式

${\displaystyle (p_{1}(x)u')'+q_{1}(x)u=0\,}$

および

${\displaystyle (p_{2}(x)v')'+q_{2}(x)v=0\,}$

の解とする。このとき、v(x) ≠ 0 であるような全ての x に対して、次の等式が成立する。

${\displaystyle \left({\frac {u}{v}}(p_{1}u'v-p_{2}uv')\right)'=\left(q_{2}-q_{1}\right)u^{2}+\left(p_{1}-p_{2}\right)u'^{2}+p_{2}\left(u'-v'{\frac {u}{v}}\right)^{2}.}$

この等式のことを、ピコーンの等式と呼ぶ。

${\displaystyle {\begin{aligned}&\left({\frac {u}{v}}(p_{1}u'v-p_{2}uv')\right)'\\&\quad =\left(up_{1}u'-p_{2}v'u^{2}{\frac {1}{v}}\right)'\\&\quad =u'p_{1}u'+u(p_{1}u')'-(p_{2}v')'u^{2}{\frac {1}{v}}-p_{2}v'2uu'{\frac {1}{v}}+p_{2}v'u^{2}{\frac {v'}{v^{2}}}\\&\quad =p_{1}u'^{2}-2p_{2}{\frac {uu'v'}{v}}+p_{2}{\frac {u^{2}v'^{2}}{v^{2}}}+u(p_{1}u')'-(p_{2}v')'{\frac {u^{2}}{v}}\\&\quad =p_{1}u'^{2}-p_{2}u'^{2}+p_{2}u'^{2}-2p_{2}u'{\frac {uv'}{v}}+p_{2}\left({\frac {uv'}{v}}\right)^{2}-u(q_{1}u)+(q_{2}v){\frac {u^{2}}{v}}\\&\quad =\left(p_{1}-p_{2}\right)u'^{2}+p_{2}\left(u'-v'{\frac {u}{v}}\right)^{2}+\left(q_{2}-q_{1}\right)u^{2}.\end{aligned}}}$

• Picone, Mauro (1910). “Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare del secondo ordine”. Ann. Scuola Norm. Sup. Pisa 11: 1–141. https://www.numdam.org/item/?id=ASNSP_1910_1_11__A1_0. 
• Swanson, Charles A. (1975). “Picone's Identity”. Rendiconti di Matematica 8 (2): 373–397.