Another concept which dictates the numerical method chosen for solving an ODE, is that of initial and boundary conditions. To give an example, if we study white dwarf stars or neutron stars we will need to solve two coupled first-order differential equations, one for the total mass \( m \) and one for the pressure \( P \) as functions of \( \rho \) $$ \frac{dm}{dr}=4\pi r^{2}\rho (r)/c^2, $$ and $$ \frac{dP}{dr}=-\frac{Gm(r)}{r^{2}}\rho (r)/c^2. $$ where \( \rho \) is the mass-energy density. The initial conditions are dictated by the mass being zero at the center of the star, i.e., when \( r=0 \), yielding \( m(r=0)=0 \). The other condition is that the pressure vanishes at the surface of the star.

In the solution of the Schroedinger equation for a particle in a potential, we may need to apply boundary conditions as well, such as demanding continuity of the wave function and its derivative.