Page 194 - Foundations Of Differential Calculus

9. On Differential Equations 177

dy ndx cos nx
= mdx + = mdx + ndx cot nx,
y sin nx
so that

dy m
arccot − = nx.
ny dx n
If we let dx be constant and differentiate, then 1
2
2
ndxdy − ny dx d y
ndx = ,
2
2 2
2
2 2
m y dx + n y dx − 2my dx dy + dy 2
or
2 2 2 2 2
m + n y dx − 2my dx dy = −yd y.
It should be clear that although the differential equation may contain no
transcendental quantity, still the finite equation from which it originated
may contain transcendental quantities of various kinds.
296. Therefore, differential equations, whether of the first or higher order,
which contain two variables, x and y, arise from finite equations that also
express a relationship between the two variables. Indeed, given any differ-
ential equation containing these two variables x and y, there is expressed
a relationship between x and y such that y becomes a function of x. From
this we can see the nature of a differential equation. That is, if we can as-
sign to y a function of x that is indicated by the equation and is such that
when the function is substituted for y its differential is substituted for dy,
3
2
and its higher differentials for d y, d y, etc., then the resulting equation
is an identity. Integral calculus is concerned with the investigation of such
functions. It has this purpose, that given any differential equation, a func-
tion of x should be defined that is equal to the other variable y, or what
is equivalent, that a finite equation be found that contains the relationship
between x and y.
297. If, for example, the given equation is
2
y dx
2ydy − adx − + xdx =0,
a
1
This is a correction by Gerhard Kowalewski. The original edition had
2
2
ndxdy − ny dx d y
ndx = .
2
2 2
2 2
2
m y dx + n y dx − 2my dx dy