⟨ y , L y ⟩ ⟨ y , y ⟩ = ∫ a b y ( x ) 1 w ( x ) ( − d d x [ p ( x ) d y d x ] + q ( x ) y ( x ) ) d x ∫ a b w ( x ) y ( x ) 2 d x = { ∫ a b y ( x ) ( − d d x [ p ( x ) y ′ ( x ) ] ) d x } + { ∫ a b q ( x ) y ( x ) 2 d x } ∫ a b w ( x ) y ( x ) 2 d x = { − y ( x ) [ p ( x ) y ′ ( x ) ] | a b } + { ∫ a b y ′ ( x ) [ p ( x ) y ′ ( x ) ] d x } + { ∫ a b q ( x ) y ( x ) 2 d x } ∫ a b w ( x ) y ( x ) 2 d x = { − p ( x ) y ( x ) y ′ ( x ) | a b } + { ∫ a b [ p ( x ) y ′ ( x ) 2 + q ( x ) y ( x ) 2 ] d x } ∫ a b w ( x ) y ( x ) 2 d x . {\displaystyle {\begin{aligned}{\frac {\langle {y,Ly}\rangle }{\langle {y,y}\rangle }}&={\frac {\int _{a}^{b}y(x){\frac {1}{w(x)}}\left(-{\frac {d}{dx}}\left[p(x){\frac {dy}{dx}}\right]+q(x)y(x)\right)dx}{\int _{a}^{b}{w(x)y(x)^{2}}dx}}\\&={\frac {\left\{\int _{a}^{b}y(x)\left(-{\frac {d}{dx}}\left[p(x)y'(x)\right]\right)dx\right\}+\left\{\int _{a}^{b}{q(x)y(x)^{2}}\,dx\right\}}{\int _{a}^{b}{w(x)y(x)^{2}}\,dx}}\\&={\frac {\left\{\left.-y(x)\left[p(x)y'(x)\right]\right|_{a}^{b}\right\}+\left\{\int _{a}^{b}y'(x)\left[p(x)y'(x)\right]\,dx\right\}+\left\{\int _{a}^{b}{q(x)y(x)^{2}}\,dx\right\}}{\int _{a}^{b}w(x)y(x)^{2}\,dx}}\\&={\frac {\left\{\left.-p(x)y(x)y'(x)\right|_{a}^{b}\right\}+\left\{\int _{a}^{b}\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx\right\}}{\int _{a}^{b}{w(x)y(x)^{2}}\,dx}}.\end{aligned}}}