Chandrasekhar's white dwarf equation

In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as

${\displaystyle {\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)+(\varphi ^{2}-C)^{3/2}=0}$

with initial conditions

${\displaystyle \varphi (0)=1,\quad \varphi '(0)=0}$

where ${\displaystyle \varphi }$ measures the density of white dwarf, ${\displaystyle \eta }$ is the non-dimensional radial distance from the center and ${\displaystyle C}$ is a constant which is related to the density of the white dwarf at the center. The boundary ${\displaystyle \eta =\eta _{\infty }}$ of the equation is defined by the condition

${\displaystyle \varphi (\eta _{\infty })={\sqrt {C}}.}$

such that the range of ${\displaystyle \varphi }$ becomes ${\displaystyle {\sqrt {C}}\leq \varphi \leq 1}$. This condition is equivalent to saying that the density vanishes at ${\displaystyle \eta =\eta _{\infty }}$.