DIFFERENTIAL EQUATIONS 
347 
a given slope (A = 2, say) are given by the equation 
V - 2/o = 2 (® - ®o)> or V = + (2/ 0 - 2ab). 
Thus the two arbitrary constants cc 0 and y 0 are together equivalent to 
but a single arbitrary constant, 
b = y 0 — 2x 0 . 
Example. Consider the differential equation 
dll _ „ 
dx 
r = Va? 2 + y 2 . 
Let it be required to find approximately where the axis of x is 
cut by the solution which cuts the axis of y one unit above the 
origin. 
The student should make an accurate drawing on squared paper, 
taking 10 cm. as the unit of length and making x k+l — x k — y 1 ^ 
(i.e. 1 cm. long). 
Simultaneous Differential Equations. A simultaneous system of 
the form 
(2) 
dy 
dx 
= y, z), 
dz . , \ 
— = $(®, y, z) 
can be treated in a similar manner. Let V be a region of space, at 
every point (x, y, z) of which the functions F and d> are continuous. 
Draw through (a*, y, z) a line whose direction components are 1, 
F(x, y, z), i>(a, y, z), and lay off a short vector along this line. A 
curve, 
(3) y = /(&), z = 4>(x) t 
which, at each of its points, is tangent to the vector pertaining to 
that point, will represent a solution of the given system (2). 
Starting at any point (x 0 , y 0 , z 0 ) of V, we can construct a broken 
line as in the earlier case, laying off first a short distance on the 
vector at (aj 0 , y 0 , %o)- From the end, (x 1} y 1} z^, of this line lay off 
a short distance on the vector pertaining to (aq, y u ^j) ; and con 
tinue in this way. The broken line thus formed will approach a 
limiting curve, (3), which represents a solution of (2), provided 
F(x, y, z) and <I>(a;, y, z) admit first partial derivatives with respect 
to y and z, which are continuous throughout V. 
The extension to the case of a simultaneous system of n equations 
in n dependent variables :