\[\begin{align} \theta_{1} - \theta_{2} &= \frac{L}{2EI}(M_{1} + M_{2}) \\ \delta_{1} - \delta_{2} + \theta_{1}L &= \frac{L^{2}}{6EI}(2M_{1} + M_{2}) \end{align}\]
\[\begin{align} M &= \kappa EI \\ \kappa &= \frac{M}{EI} \\ \kappa &= \frac{1}{EI}\bigg(M_{1} + \frac{x}{L}(M_{2}-M_{1})\bigg) \end{align}\]
\[\begin{align} \theta &= -\int \kappa\,dx \\ &= -\frac{1}{EI} \int M_{1} + \frac{x}{L}(M_{2}-M_{1})\,dx \\ &= -\frac{1}{EI} \bigg[M_{1}x + \frac{x^{2}}{2L}(M_{2}-M_{1}) + A] \end{align}\]
where \(x=0\), \(\theta = \theta_{1}\)
\[\begin{align} \theta_{1} &= -\frac{A}{EI} \\ A &= -EI\theta_{1} \end{align}\]
where \(x=L\), \(\theta = \theta_{2}\)
\[\begin{align} \theta_{2} &= -\frac{1}{EI}\bigg[M_{1}L + \frac{L^{2}}{2L}(M_{2}-M_{1}) - EI\theta_{1}\bigg] \\ \theta_{2} &= -\frac{1}{EI}\bigg[M_{1}L + \frac{L}{2}(M_{2} - M_{1})\bigg] + \theta_{1} \\ \theta_{1} - \theta_{2} &= \frac{1}{EI}\bigg[M_{1}\frac{L}{2} + M_{2}\frac{L}{2}\bigg] \\ \theta_{1} - \theta_{2} &= \frac{L}{2EI}(M_{1} + M_{2}) \end{align}\]
\[\begin{align} \delta &= \int \theta \, dx \\ &= -\frac{1}{EI} \int M_{1}x + \frac{x^{2}}{2L}(M_{2}-M_{1}) - EI\theta_{1}\, dx \\ &= -\frac{1}{EI} \bigg[ \frac{M_{1}x^{2}}{2} + \frac{x^{3}}{6L}(M_{2} - M_{1}) - EI\theta_{1}x + B\bigg] \end{align}\]
where \(x = 0\), \(\delta = \delta_{1}\)
\[\begin{align} \delta_{1} &= -\frac{1}{EI} B \\ B &= -EI \delta_{1} \end{align}\]
where \(x = L\), \(\delta = \delta_{2}\)
\[\begin{align} \delta_{2} &= -\frac{1}{EI}\bigg[\frac{M_{1}L^{2}}{2} + \frac{L^{3}}{6L}(M_{2} - M_{1}) - EI\theta_{1}L - EI\delta_{1}\bigg] \\ \delta_{2} &= -\frac{1}{EI}\bigg[\frac{M_{1}L^{2}}{3} + \frac{M_{2}L^{2}}{6}\bigg] + \theta_{1}L + \delta_{1} \\ \delta_{1} - \delta_{2} + \theta_{1}L &= \frac{L^{2}}{6EI}(2M_{1} + M_{2}) \end{align}\]