SEMANTICS OF THE DIFFERENTIAL CALCULUS 595
equal to the sum of the partial differentials of the first order if we neglect terms of higher orders, whose values are indefinitely small quantities relative to
the first order differentials. In symbols, df^dj+dyf^l -^ P^+It- W- ^n
words, the total differential of is found by finding the partial derivatives
with respect to x and y, multiplying them respectively by dx and dy, and adding.
2. DIFFERENTIAL EQUATIONS
A natural development of the invention of the calculus was the introduction of differential equations. Differential equations differ from the ordinary equations of mathematics in that in addition to variables and constants they contain also derivatives of one or more of the variables involved. Differential equations are of extreme importance, and arise in many problems. Newton solved his first differential equation in 1676 by the use of an infinite series, eleven years after his discovery of the calculus in 1665. Leibnitz solved his first differential equation in 1693, the year in which Newton first published his results. From this date on, progress in the development and application of differential equations was very rapid, and today the subject of differential equations occupies in the general field of mathematics a central position from which important and useful lines of development flow in many different directions.
To integrate or solve a differential equation means, analytically, to find all the functions which satisfy the equation. In geometry, it means to find all the curves which have the property expressed by the equation. In mechanics it means to find all the motions that may possibly result from a given set of forces ,. The degree of the differential equation is defined as the degree of the derivative of the highest order which enters the equation. The order of the differential equations is the order of the highest derivative it contains.
Equations in * and y, of the first degree in y and its derivatives with respect to x, y', y"., are called linear equations. The main equations of physics are linear differential equations of the second order, since y, the primitive function, y', the first derivative, and y", the second derivative, appear only in the first
degree. For instance the equation
when X represents a function of * alone is such an equation. It is linear, or of the first degree, because the second derivative, y", appears only to the first degree. It is of the second order because that is the highest order derivative in the equation. As we may recall, the derivative of a function gives us the rate of change of the function when we give successive values to the independent variable. When we study the rate of change of the rate of change of our function, we study the rate of change of the first derivative which expresses the rate of change of the function, whence we obtain the derivative of the second order, and so on. If we equate our derivatives to zero, or choose a value of the variable for which our derivative becomes zero, the rate of change of our function