Page 187 - Foundations Of Differential Calculus
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170 9. On Differential Equations
and if this is multiplied by y,wehave
3 2 2
y dy + x ydx = axy dy + ay dx.
3 3
If we substitute for y its value 3axy − x , we obtain the new equation
3 2 2
2ax dy − x dy + x ydx = ay dx.
3
If we multiply this equation again by y and then substitute for y its value,
we have
3
2
3
2 2
2
2axy dy − x ydy + x y dx =3a xy dx − ax dx.
In general, if P, Q, R represent any functions of x and y, and if the differ-
ential equation is multiplied by P, then
2 2
Py dy + Px dx = aPx dy + aPy dx.
3
3
Now, since x + y − 3axy = 0, we also have
3 3
x + y − 3axy (Qdx + Rdy)=0.
When these equations are added to each other we obtain a general differ-
ential equation that arises from the given finite equation
3
2
3
Py dy − aPx dy + Rx dy + Ry dy − 3aRxy dy
2 3 3
+ Px dx − aPy dx + Qx dx + Qy dx − 3aQxy dx =0.
287. It is possible to find an infinite number of differential equations
through differentiation from the same finite equation, since before differen-
tiation the equation can be multiplied or divided by an arbitrary quantity.
Thus, if P is any function of x and y, so that dP = pdx + qdy, and if
the finite equation is multiplied by P and then differentiated, we obtain a
general differential equation that takes on an infinite number of forms in-
sofar as P takes on one or another function. This multiplicity is increased
infinitely if this equation is added to the original finite equation multiplied
by the formula Qdx+Rdy, where Q and R can be any functions of x and y.
Although in all of these equations the relation between dy and dx remains,
and this is determined by the differential of the function y in the original
finite equation, nevertheless, the differential of y could be determined by
other finite equations. The reason for this is better explained in integral
calculus.
288. Not only can we obtain an innumerable number of equations from
a single finite equation, but we can also find an infinite number of finite
equations that lead to the same differential equation. For instance, these