20.6 First order equations in more dimensions

The procedures of the previous subsections extend in a logical way to more dimensions. If the independent variables are $x_1,x_2,\ldots,x_n$, the first order quasi-linear partial differential equation takes the form

$$\fbox{$\displaystyle a_1 u_{x_1} + a_2 u_{x_2} + \ldots + a_n u_{x_n} = d $} %$$ (20.6)

where the $a_n$ and $d$ may depend on $x_1,x_2,\ldots,x_n$ and $u$.

The characteristic equations can now be found from the ratios

$$\fbox{$\displaystyle {\rm d}x_1 : {\rm d}x_2 : \ldots : {\rm d}x_n : {\rm d}u = a_1 : a_2 : \ldots : a_n : d $} %$$ (20.7)

After solving $n$ different ordinary differential equations from this set, the integration constant of one of them, call it $C_n$ can be taken to be a general $n-1$-parameter function of the others,

$$C_n=C_n(C_1,C_2,\ldots,C_{n-1})$$

and then substituting for $C_1,C_2,\ldots,C_{n-1}$ from the other ordinary differential equation, an expression for $u$ results involving one still undetermined, $n-1$ parameter function $C_n$.

To find this remaining undetermined function, plug in whatever initial condition is given, renotate the parameters of $C_n$ to $\alpha_1,\alpha_2,\ldots$ and express everything in terms of them to find function $C_n(\alpha_1,\alpha_2,\ldots)$.