DEFINITION
An equation involving the derivative(s) of dependent
variable y w.r.t. independent variable x or equation of
dependent variables with respect to independent variables
involving derivative is called a differential equation.
A differential equation involving derivatives with
respect to only one independent variable is called
ordinary differential equation.
ORDER AND DEGREE OF A DIFFERENTIAL
EQUATION
z The order of the highest order derivative occurring
in the given differential equation is called the order
of the differential equation.
z The power of the highest order derivative occurring
in the differential equation is called the degree of
the differential equation.
Note : Order and degree (if defined) of a differential
equation are always positive integers.
SOLUTION OF A DIFFERENTIAL EQUATION
A relation between the independent and dependent
variables free from derivatives satisfying the given
differential equation is called a solution of the given
differential equation.
GENERAL AND PARTICULAR SOLUTIONS OF A
DIFFERENTIAL EQUATION
z The general solution of a differential equation is the
the relation between the variables (not involving
the differential coefficients) satisfying the given
differential equation and containing as many
arbitrary constants as its order is.
HIGHLIGHTSCLASS XII Series 6
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VSA ––22--
SA 22--33
LA --11--Differential Equations
z The solution of the differential equation for
particular values of one or more of the arbitrary
constants is called a particular solution of the given
differential equation.
FORMATION OF A DIFFERENTIAL EQUATION
WHOSE GENERAL SOLUTION IS GIVEN
Suppose an equation of a family of curves contains
n arbitrary constants (called parameters).
Then, we obtain its differential equation, using following
steps :
Step I : Differentiate the equation of the given family of
curves n times to get n more equations.
Step II : Eliminate n constants, using these (n + 1)
equations.
This gives us the required differential equation of order
n.
METHODS OF SOLVING FIRST ORDER, FIRST
DEGREE DIFFERENTIAL EQUATIONS
(i) If the equation is dy
dx=fx(), then y = ∫^ f(x)dx + C^
is the solution.
(ii) Variable separable : If the given differential
equation can be expressed in the form f(x)dx = g(y)dy,
then ∫ f(x)dx = ∫ g(y)dy + C is the solution.