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 $$ t=b $$.

$$\displaystyle \frac {d^2 x}{dt^2} + p(t) \frac {d x}{dt} + q(t) x(t) = f(t)$$ with $$\displaystyle a_0x(a) +a_1x'(a)= g$$ and  $$\displaystyle b_0x(b) +b_1x'(b)= h$$ on the interval $$\displaystyle I_{ab} = \{x | a \leq t \leq b \} $$



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