Page:Chandrasekhar - On the decay of plane shock waves.djvu/25

constructed. For this purpose we require the general integral of equation (16). As is well known (cf. A.R. Forsyth, Differential Equations, pp. 375-380) the general integral can be readily written down in terms of a complete integral, i.e., an integral which contains as many constants as independent variables. Writing the complete integral of equation (16) in the form (cf. eq. (17))

(38)$\phi ={\frac {a_{1}+x}{a_{2}+t}}$

where a₁ and a₂ are two arbitrary constants, the general integral of equation (16) can be expressed as the eliminant between the equations

(36)

$\phi \chi (a)+t=a+x$	,
$\phi {\frac {d\chi }{da}}=1$	,

where χ is any arbitrary function of a. It is now evident that with the solution in the form (36) we can make φ satisfy any arbitrary distribution at time t = 0. Alternatively we may say that the distribution of c (or equivalently u) at time t = 0 will determine χ(a) thus making the solution determinate. In this fashion the most general form of shock pulses under the assumptions made in §1 can be constructed. In a later report we propose to give examples of shock pulses belonging to this more general class.

S. Chandrasekhar

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