Boundary Value Problems/ODE BVPs

Begin with second order DEs, ${\displaystyle x''=f(t,x,x')}$, with conditions on the solution at ${\displaystyle t=a}$ and ${\displaystyle t=b}$.
${\displaystyle \displaystyle {\frac {d^{2}x}{dt^{2}}}+p(t){\frac {dx}{dt}}+q(t)x(t)=f(t)}$ with ${\displaystyle \displaystyle a_{0}x(a)+a_{1}x'(a)=g}$ and ${\displaystyle \displaystyle b_{0}x(b)+b_{1}x'(b)=h}$ on the interval ${\displaystyle \displaystyle I_{ab}=\{x|a\leq t\leq b\}}$
${\displaystyle \displaystyle {\frac {d^{2}x}{dt^{2}}}+4{\frac {dx}{dt}}+2x(t)=f(t)}$ with ${\displaystyle \displaystyle x(0)=0}$ and ${\displaystyle \displaystyle x(1)=0}$ on the interval ${\displaystyle \displaystyle I_{ab}=\{x|0\leq t\leq 1\}}$