I'm studying finance, looking through Dixit & Pindyck's Investment under uncertainty. They have this homogeneous 2nd order ODE, where F(V) is a function of V:
$$\frac{1}{2}\sigma^2 V^2 F''(V) + (\rho-\delta)VF'(V) - \rho F = 0 \tag{1}$$
F(V) must satisfy three boundary conditions:
$$F(0) = 0 \tag{2}$$
$$F(V^*) = V^*-I \tag{3}$$
$$F'(V^*) = 1 \tag{4}$$
From high school, I remember that a 2nd order ODE can be
$$c_1e^{r_1x} + c_2e^{r_2x} \tag{5}$$
$$c_1e^{rx} + xc_2e^{rx} \tag{6}$$
$$e^{\alpha x}(c_1sin(\beta x) + c_2cos(\beta x)) \tag{7}$$
Where (5) is for real, distinct roots, (6) is for real, repeated roots and (7) is for complex roots $\alpha \pm \beta i$.
(1) is a homongeneous, 2nd order ODE, so I would assume the solution would be in the form (5), (6) or (7). However, without further ado, the authors of my finance textbook write
To satisfy the boundary condition (2), the solution must take the form
$$F(V) = AV^{\beta _1}$$
This solution doesn't look like neither (5), (6) nor (7). Like, where's the $e$? Why is the solution of this homogeneous, 2nd order ODE not in the expected form?
The solutions you remember are for a constant coefficient second order linear homogeneous equation. In your equations the coefficients are not constant, and your equation is in fact a Cauchy-Euler equation, where the solutions are linear combinations of powers of $V$ (or a bit worse in some cases, depending on the coefficients).