I am trying to solve a ODE system with the following equations: $$ \frac{\partial{\beta}}{\partial{t}}=k\beta-\frac{\sigma^2\beta^2}{2}+\frac{\delta\beta+u}{\delta\beta+\eta+u} $$ $$ \frac{\partial\alpha}{\partial{t}}=-k\theta\beta $$ where the boundary condition:$$\alpha(t)=0,\beta(t)=0$$ and $$k,\sigma,\delta,u,\eta,\theta$$ are given parameters.
In my opinion,this is not a intial value problem, so I cannot use the solver ode45 directly. bvp4c also doesn't work.
Any suggestions?