微分演算子の交換則

微分演算子 ${\displaystyle f\left(D\right)}$ ， ${\displaystyle g\left(D\right)}$ に対して

${\displaystyle f\left(D\right)g\left(D\right)=g\left(D\right)f\left(D\right)}$

が成り立つ．すなわち，交換則が成り立つ．

■具体的な解説

${\displaystyle f\left(D\right)=D+C_1}$　･･････(1)

${\displaystyle g\left(D\right)=D+C_2}$　･･････(2)

とする．

${\displaystyle \left\{f\left(D\right)g\left(D\right)\right\}y=f\left(D\right)\left\{g\left(D\right)y\right\}}$

(微分演算子の積を参照)

${\displaystyle =\left(D+C_1\right)\left\{g\left(D\right)y\right\}}$

${\displaystyle =D\left\{g\left(D\right)y\right\}+C_1\left\{g\left(D\right)y\right\}}$

(微分演算子の和を参照)

${\displaystyle =D\left\{\left(D+C_2\right)y\right\}+C_1\left\{\left(D+C_2\right)y\right\}}$

${\displaystyle =\left(D^2+C_{2}D\right)y+\left(C_{1}D+C_1{C}_2\right)y}$

${\displaystyle =D^{2}y+C_{2}Dy+C_{1}Dy+C_1{C}_{2}y}$

${\displaystyle =D^{2}y+C_{1}Dy+C_{2}Dy+C_1{C}_{2}y}$

${\displaystyle =\left(D^2+C_{1}D\right)y+\left(C_{2}D+C_1{C}_2\right)y}$

${\displaystyle =D\left\{\left(D+C_1\right)y\right\}+C_2\left\{\left(D+C_1\right)y\right\}}$

${\displaystyle =D\left\{f\left(D\right)y\right\}+C_1\left\{f\left(D\right)y\right\}}$

${\displaystyle =\left(D+C_2\right)\left\{f\left(D\right)y\right\}}$

${\displaystyle =g\left(D\right)\left\{f\left(D\right)y\right\}}$

${\displaystyle =\left\{g\left(D\right)f\left(D\right)\right\}y}$

よって

${\displaystyle f\left(D\right)g\left(D\right)=g\left(D\right)f\left(D\right)}$

が成り立つ．

 

ホーム>>カテゴリー別分類>>微分>>微分方程式>>微分演算子の交換則

学生スタッフ作成
最終更新日： 2023年6月26日