Page 131 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes  The degree of a differential equation is the degree of the highest derivative occurring in it,
after the equation has been expressed in a form free from radials and fractions so far as the
derivatives are concerned.
 Any relation between the dependent and independent variables, when substituted in the
differential equation, reduces it to an identity is called a solution or integral of the
differential equation.
 The solution of a differential equation, in which the number of arbitrary constants is equal
to the order of the differential equation is called the general solution or complete solution
or the complete primitive.

 The solution obtained from the general solution by giving particular values to the arbitrary
constants, is called a particular solution of the differential equation.
 A solution or integral of a differential equation is a relation between the variables, by
means of which and the derivatives obtained therefore, the equation is satisfied.
dy
x
y
0
 A differential equation of first order and first degree is of the form  f ( , )  which
dx
is sometimes written as Mdx + Ndy = 0. Where M and N are functions of x and y or constants
and f is some known function of x.
 A first order first degree equation of the form M(x, y)dx + N(x, y) dy = 0 is said to be exact
if its left hand side quantity is the exact differential of some function u(x, y).

9.6 Keywords

Complete Primitive: The solution of a differential equation, in which the number of arbitrary
constants is equal to the order of the differential equation is called the general solution or
complete solution or the complete primitive.
Degree: The degree of a differential equation is the degree of the highest derivative occurring in
it, after the equation has been expressed in a form free from radials and fractions so far as the
derivatives are concerned.
Differential Equation: A differential equation involves independent variables, dependent
variables, their derivatives and constants.
Order: The order of a differential equation is the order of highest derivative appearing in the
equation.
Particular Solution: The solution obtained from the general solution by giving particular
values to the arbitrary constants, is called a particular solution of the differential equation.
9.7 Review Questions


1. Solve the differential equation x 2   1y  dx y 2   1x  dy  0 .

2. Solve the differential equation  1 x 2  1 y dx  xy  1 y dy .

dy  2 dy 
3. Solve the differential equation y  x  a y  .

dx  dx 
dy 
4. Solve the differential equation x  coty  0 if y  when x  2 .
dx 4

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