チェビシェフの微分方程式

チェビシェフの微分方程式

(1)

\[ \left(1-x^{2}\right)T_{n}''(x)-xT_{n}'(x)+n^{2}T_{n}(x)=0 \]

(2)

\[ \left(1-x^{2}\right)U_{n}''(x)-3xU_{n}'(x)+n(n+2)U_{n}(x)=0 \]

(1)

\begin{align*} \left(1-x^{2}\right)T_{n}''(x)-xT_{n}'(x)+n^{2}T_{n} & =\left(1-x^{2}\right)\left(T_{n}''(x)+\frac{-x}{1-x^{2}}T_{n}'(x)+\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\exp\left(-\int^{x}\frac{-t}{1-t^{2}}dt\right)\frac{d}{dx}\left\{ \exp\left(\int^{x}\frac{-t}{1-t^{2}}dt\right)T_{n}'(x)\right\} +\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\exp\left(-\frac{1}{2}\log(1-x^{2})\right)\frac{d}{dx}\left\{ \exp\left(\frac{1}{2}\log(1-x^{2})\right)T_{n}'(x)\right\} +\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\frac{1}{\sqrt{1-x^{2}}}\frac{d}{dx}\left\{ \sqrt{1-x^{2}}T_{n}'(x)\right\} +\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\frac{1}{\sqrt{1-x^{2}}}\frac{d}{dx}\left\{ \sqrt{1-x^{2}}\frac{d}{dx}\cos\left(n\cos^{\circ}x\right)\right\} +\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\frac{1}{\sqrt{1-x^{2}}}\frac{d}{dx}\left\{ \sqrt{1-x^{2}}\sin\left(n\cos^{\circ}x\right)\frac{n}{\sqrt{1-x^{2}}}\right\} +\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\frac{n}{\sqrt{1-x^{2}}}\frac{d}{dx}\sin\left(n\cos^{\circ}x\right)+\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(\frac{n}{\sqrt{1-x^{2}}}\cos\left(n\cos^{\circ}x\right)\frac{-n}{\sqrt{1-x^{2}}}+\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =\left(1-x^{2}\right)\left(-\frac{n^{2}}{1-x^{2}}T_{n}(x)+\frac{n^{2}}{1-x^{2}}T_{n}(x)\right)\\ & =0 \end{align*}

(2)

\begin{align*} \left(1-x^{2}\right)U_{n}''(x)-3xU_{n}'(x)+n(n+2)U_{n}(x) & =\frac{1}{n+1}\left\{ \left(1-x^{2}\right)T_{n+1}'''(x)-3xT_{n+1}''(x)+n(n+2)T_{n+1}'(x)\right\} \qquad,\qquad nU_{n-1}(x)=T_{n}'(x)\\ & =\frac{1}{n+1}\left\{ \frac{d}{dx}\left(\left(1-x^{2}\right)T_{n+1}''(x)\right)+2xT_{n+1}''(x)-3xT_{n+1}''(x)+n(n+2)T_{n+1}'(x)\right\} \\ & =\frac{1}{n+1}\left\{ \frac{d}{dx}\left(\left(1-x^{2}\right)T_{n+1}''(x)\right)-\frac{d}{dx}\left(xT_{n+1}'(x)\right)+T_{n+1}'(x)+n(n+2)T_{n+1}'(x)\right\} \\ & =\frac{1}{n+1}\left\{ \frac{d}{dx}\left(\left(1-x^{2}\right)T_{n+1}''(x)\right)-\frac{d}{dx}\left(xT_{n+1}'(x)\right)+(n+1)^{2}\frac{d}{dx}T_{n+1}(x)\right\} \\ & =\frac{1}{n+1}\frac{d}{dx}\left\{ \left(1-x^{2}\right)T_{n+1}''(x)-xT_{n+1}'(x)+(n+1)^{2}T_{n+1}(x)\right\} \\ & =0 \end{align*}

ページ情報

タイトル

チェビシェフの微分方程式

URL

https://www.nomuramath.com/br9x8q6f/

SNSボタン