Boundary Value Problems/ODE BVPs

Two point BVPs for an ODE

Begin with second order DEs, $x''=f(t,x,x')$, with conditions on the solution at $t=a$ and $t=b$.

$\displaystyle {\frac {d^{2}x}{dt^{2}}}+p(t){\frac {dx}{dt}}+q(t)x(t)=f(t)$ with $\displaystyle a_{0}x(a)+a_{1}x'(a)=g$ and $\displaystyle b_{0}x(b)+b_{1}x'(b)=h$ on the interval $\displaystyle I_{ab}=\{x|a\leq t\leq b\}$

Example

$\displaystyle {\frac {d^{2}x}{dt^{2}}}+4{\frac {dx}{dt}}+2x(t)=f(t)$ with $\displaystyle x(0)=0$ and $\displaystyle x(1)=0$ on the interval $\displaystyle I_{ab}=\{x|0\leq t\leq 1\}$