1) The order of a differential equation is defined as the order of the highest derivative term involved.
2) The degree of a differential equation is defined as the degree of the highest derivative term after making the equation free of radicals and fractions involving derivatives.
3) A linear differential equation is one where each term is only of the first degree and there are no multiplied derivative terms. Any equation that does not satisfy these is non-linear.
2. Order of a Differential Equation
The order of the highest order derivative
involved in a differential equation is called the
order of the differential equation.
3. Degree of a Differential Equation
The degree of the highest derivative which occurs in it, after the
differential equation has been made free from radicals and
fractions as far as the derivatives are concerned.
Note that the above definition of degree does not require
variables x, t, u etc. to be free from radicals and fractions.
4. Linear and Non-linear Equations
β’Linear-if (i) every dependent variable and every derivative involved
occurs in the first degree only and
(ii) no products of dependent variables and/or derivatives occur.
β’Non-linear- if an equation is not linear
5. Consider π¦ = π₯
!"
!#
+
$
!"
!#
π = π
π π
π π
+
π
π π
π π
β π.
π π
π π
= π
π π
π π
π
+ π by multiplying by
π π
π π
to
get rid of the fraction
π
π π
π π
from the previous equation.
Then order=1
Degree=2 i.e., degree of
π π
π π
in the term π
π π
π π
π
6. Consider
!*"
!#*
%
+ π₯
!"
!#
&
+ π¦ = π₯'
β’The equation is already free from radicals and fractions as far
as the derivatives are concerned.
β’So order=order of the highest order derivative=2 since
derivatives of only the first and second order are present here.
β’Degree= degree of the highest order derivative=3
7. Consider
!*"
!#*
+
,
= π¦ +
!"
!#
+
*
The order= order of the highest order derivative=2
The equation is not from from radicals of derivatives.
So to find degree, let us transform the equation as π π π
π π π
π
=
π +
π π
π π
π
by raising both sides to 6th power.
Now the degree= degree of the 2nd order derivative=2
8. Consider π¦ + π₯
!"
!#
'
.
,
= π₯
!*"
!#*
The order= order of the highest order derivative=2
The equation is not from from radicals of derivatives.
So to find degree, let us transform the equation as π¦ + π₯
!"
!#
$ %
=
π₯& !!"
!#!
&
by raising both sides to 3rd power.
Now the degree= degree of the 2nd order derivative=3
