User:Dross/Math

$$\begin{align} \frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle} &=\frac{\int_a^b y(x)\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + q(x)y(x)\right)dx}{\int_a^b{w(x)y(x)^2}dx}\\ &= \frac{ \left \{ \int_a^b y(x)\left(-\frac{d}{dx}\left[p(x)y'(x)\right]\right) dx \right \}+ \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b{w(x)y(x)^2} \, dx} \\ &= \frac{ \left \{\left. -y(x)\left[p(x)y'(x)\right] \right |_a^b \right \} + \left \{\int_a^b y'(x)\left[p(x)y'(x)\right] \, dx \right \} + \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b w(x)y(x)^2 \, dx}\\ &= \frac{ \left \{ \left. -p(x)y(x)y'(x) \right |_a^b \right \} + \left \{ \int_a^b \left [p(x)y'(x)^2 + q(x)y(x)^2 \right] \, dx \right \} } {\int_a^b{w(x)y(x)^2} \, dx}. \end{align}$$