### Differential equations: Separation of variables

### Differentials

A very suggestive and convenient way of working with derivatives of functions emerges when we make the transition to differentials.

Differentials

A **differential** is an expression of the form #f(x)\, \dd g(x)# where #f# and #g# are functions of #x#. It is neither a number nor a function, but an expression indicating an "infinitely small" (infinitesimal) change depending on the change in #g(x)#. By definition, the term satisfies the law \[f(x)\, \dd g(x) =f(x)\cdot g'(x)\,\dd x\]

It is not necessary to put the #\dd# part at the back; we agree:

\[ \left(\dd f(x)\right)\cdot g(x)= g(x)\, \dd f(x) \]

An equation in which the terms are differentials is called an **equation in differential form.**

Often, the differential #1\,\dd x# is written as #\dd x#, and #0\,\dd x# as #0#.

By use of differentials the differentiation laws are simple to formulate:

Rules for differentiation in terms of differentials

- product rule: #\dd\left(f(x)\cdot g(x) \right)= \left( \dd f(x)\right)\cdot g(x)+f(x)\cdot\dd\left( g(x) \right)#
- extended summation rule: #\dd\left(a\cdot f(x)+b\cdot g(x) \right)= a\,\dd f(x)+b\,\dd g(x) #
- chain rule: #\dd f\left((g(x) \right) = f'\left( g(x)\right)\,\dd g(x)#
- quotient rule: #\dd \dfrac{f(x)}{g(x)} = \frac{\dd f(x)}{g(x)} -\frac{ f(x)\,\dd g(x)}{g(x)^2}#

Implicit differentiation is also easy to formulate in terms of differentials:

Implicit differentiation in terms of differentials

Let #F(x,y)# be a function rule in #x# and #y#. If #C# is a constant and #F(x,y) = C#, then we have #\dd F(x,y)=0#.

By way of example, we consider the circle around the origin with radius #1# in #x,y#-plane. This is determined by the equation \[x^2+y^2=1\]

Left and right of the equality sign, we take the differential, and find #2x\,\dd x+2y\,\dd y = 0#, which, after division by #2#, gives

\[x\,\dd x+y\,\dd y = 0\]

Each circle around the origin is a solution of this equation in differential form.

Also integrals can be formulated in terms of differentials:

Integration in terms of differentials

If #f(x)\,\dd x=\dd\, F(x)#, then #F(x) =\int f(x)\,\dd x#.

The constants of integration left and right of the equality sign give no new solutions and may therefore be accommodated in a single integration constant. In the formula #F(x) =\int f(x)\,\dd x# this happened implicitly, as the integral on the right hand side indicates a function up to a constant.

**Pass Your Math**independent of your university. See pricing and more.

Or visit omptest.org if jou are taking an OMPT exam.