先端集中荷重の場合
\[
M(x)=Wx
\\
\theta(x)=\dfrac{W}{2EI}x^2+C_{1}
\\
\delta(x)=\dfrac{W}{6EI}x^3+C_{1}x+C_{2}
\\
\]
となり、境界条件より
\[
\theta(L)=\dfrac{WL^2}{2EI}+C_{1}=0
\\
C_{1}=-\dfrac{WL^2}{2EI}
\\
\delta(L)=\dfrac{WL^3}{6EI}-\dfrac{WL^3}{2EI}+C_{2}=0
\\
C_{2}=\dfrac{WL^3}{3EI}
\]
よって
\[
\theta(x)=\dfrac{W}{2EI}x^2-\dfrac{WL^2}{2EI}
\\
\delta(x)=\dfrac{W}{6EI}x^3-\dfrac{WL^2}{2EI}x+\dfrac{WL^3}{3EI}
\]
また、最大たわみ量\(\delta_{max}\)は
\[
\begin{vmatrix}\delta(x)\end{vmatrix}_{max}=\begin{vmatrix}\delta(0)\end{vmatrix}=
\dfrac{WL^3}{3EI}
\]
\[
M(x)=\dfrac{w}{2}x^2
\\
\theta(x)=\dfrac{w}{6}x^3+C_{1}
\\
\delta(x)=\dfrac{w}{24}x^4+C_{1}x+C_{2}
\]
となり、境界条件より
\[
\theta(L)=\dfrac{w}{6}L^3+C_{1}=0
\\
C_{1}=-\dfrac{w}{6}L^3
\\
\delta(L)=\dfrac{w}{24}L^4-\dfrac{w}{6}L^4+C_{2}=0
\\
C_{2}=\dfrac{w}{8}L^4
\]
よって
\[
\theta(x)=\dfrac{w}{6}x^3-\dfrac{w}{6}L^3
\\
\delta(x)=\dfrac{w}{24}x^4-\dfrac{w}{6}L^3x+\dfrac{w}{8}L^4
\]
また、最大たわみ量\(\delta_{max}\)は
\[
\delta_{max}=
\begin{vmatrix}\delta(0)\end{vmatrix}=\dfrac{w}{8}L^4=\dfrac{W}{8}L^3
\\
※W=wl
\]