082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

536 Chapter 19 Ordinary differential equations I
y R
y S
+
–
i
R
C
y C
Figure19.1
AnRC charging circuit.
and hence
RC dv C
+v
dt C
=v S
This isthedifferential equation which models the variation involtageacross thecapacitorwithtime.Here
v C
isthedependentvariableandt istheindependentvariable,
and when we are required to solve this differential equation we must attempt to find
v C
as a function oft.
19.2.1 Order
Differential equations which have features in common are often grouped together and
givencertainclassificationsanditisusuallythecasethatappropriatemethodsofsolution
depend upon the classifications. Someimportantterminology isnow given.
Theorder ofadifferential equation isthe orderofits highestderivative.
Example19.1 State the orderof
(a) d2 y
dx 2 + dy
dx =x
dx
(b)
dt = (xt)5
Solution (a) The highest derivative is d2 y
dx2, a second derivative. The order istherefore two.
(b) The only derivative appearing is dx , a first derivative. The order istherefore one.
dt
19.2.2 Linearity
Recall that in a differential equation such as dy
dx +3y =x2 , the independent variable is
x and the dependent variable isy.
Adifferential equation issaidtobelinearif:
(1) the dependent variable and itsderivatives occur tothe first power only,
(2) thereare no products involving the dependent variable with its derivatives, and
(3) there are no non-linear functions of the dependent variable such as sine, exponential,etc.
Ifanequationisnotlinear,thenitissaidtobenon-linear.Notefrom(2)thataproduct
of terms involving the dependent variable such asy dy is non-linear. Note from (3) that
dx
the existence of termssuch asy 2 , siny and e y causes anequation tobe non-linear.