Flashcards: Differential Equations | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
DIFFERENTIAL EQUATION 
An equation involving dependent and independent variables and derivatives of dependent variables is 
called a Differential Equation. It is expressed as : 
?? (?? , ?? , ?? , ?? , … ,
????
????
,
?? 2
?? ?? ?? 2
, … ) = 0
????
????
=
?? ( ?? )
?? ( ?? )
,
?? 2
?? ?? ?? 2
+ ?? ????
????
+ ?? 2
= 0
 
All are examples of differential equations 
  
Page 2


 
 
 
DIFFERENTIAL EQUATION 
An equation involving dependent and independent variables and derivatives of dependent variables is 
called a Differential Equation. It is expressed as : 
?? (?? , ?? , ?? , ?? , … ,
????
????
,
?? 2
?? ?? ?? 2
, … ) = 0
????
????
=
?? ( ?? )
?? ( ?? )
,
?? 2
?? ?? ?? 2
+ ?? ????
????
+ ?? 2
= 0
 
All are examples of differential equations 
  
 
 
 
Types of Differential Equations 
(1) Ordinary differential equation : A differential equation involving only one independent variable 
and derivatives w.r.t. this single independent variable is called an ordinary differential equation. 
For example : 
????
?? ?? = ?? + ?? ,
?? 4
?? ?? ?? 4
+
?? 2
?? ?? ?? 2
(
????
????
)
2
= ?? ?? 
  
Page 3


 
 
 
DIFFERENTIAL EQUATION 
An equation involving dependent and independent variables and derivatives of dependent variables is 
called a Differential Equation. It is expressed as : 
?? (?? , ?? , ?? , ?? , … ,
????
????
,
?? 2
?? ?? ?? 2
, … ) = 0
????
????
=
?? ( ?? )
?? ( ?? )
,
?? 2
?? ?? ?? 2
+ ?? ????
????
+ ?? 2
= 0
 
All are examples of differential equations 
  
 
 
 
Types of Differential Equations 
(1) Ordinary differential equation : A differential equation involving only one independent variable 
and derivatives w.r.t. this single independent variable is called an ordinary differential equation. 
For example : 
????
?? ?? = ?? + ?? ,
?? 4
?? ?? ?? 4
+
?? 2
?? ?? ?? 2
(
????
????
)
2
= ?? ?? 
  
 
 
 
 
 
Types of Differential Equations 
(2) Partial Differential Equation : A differential equation involving partial derivatives w.r.t. more 
than one independent variables is called a partial differential equation. 
For example : 
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
= 0 
  
Page 4


 
 
 
DIFFERENTIAL EQUATION 
An equation involving dependent and independent variables and derivatives of dependent variables is 
called a Differential Equation. It is expressed as : 
?? (?? , ?? , ?? , ?? , … ,
????
????
,
?? 2
?? ?? ?? 2
, … ) = 0
????
????
=
?? ( ?? )
?? ( ?? )
,
?? 2
?? ?? ?? 2
+ ?? ????
????
+ ?? 2
= 0
 
All are examples of differential equations 
  
 
 
 
Types of Differential Equations 
(1) Ordinary differential equation : A differential equation involving only one independent variable 
and derivatives w.r.t. this single independent variable is called an ordinary differential equation. 
For example : 
????
?? ?? = ?? + ?? ,
?? 4
?? ?? ?? 4
+
?? 2
?? ?? ?? 2
(
????
????
)
2
= ?? ?? 
  
 
 
 
 
 
Types of Differential Equations 
(2) Partial Differential Equation : A differential equation involving partial derivatives w.r.t. more 
than one independent variables is called a partial differential equation. 
For example : 
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
= 0 
  
 
 
 
Types of Differential Equations 
(3) Linear and non-linear differential equations : A differential equation is called linear if (i) 
every dependent variable and every derivative involved occurs in the first degree and (ii) no product of 
dependent and/or derivatives occur. A differential equation which is not linear is called non-linear 
differential equations. The general form of a ?? th 
 order linear differential equation is 
?? ?? ( ?? )
?? ?? ?? ?? ?? ?? + ?? ?? -1
( ?? )
?? ?? -1
?? ?? ?? ?? -1
+ ? + ?? 1
( ?? )
????
????
+ ?? 0
( ?? ) ?? = ?? ( ?? ) 
  
Page 5


 
 
 
DIFFERENTIAL EQUATION 
An equation involving dependent and independent variables and derivatives of dependent variables is 
called a Differential Equation. It is expressed as : 
?? (?? , ?? , ?? , ?? , … ,
????
????
,
?? 2
?? ?? ?? 2
, … ) = 0
????
????
=
?? ( ?? )
?? ( ?? )
,
?? 2
?? ?? ?? 2
+ ?? ????
????
+ ?? 2
= 0
 
All are examples of differential equations 
  
 
 
 
Types of Differential Equations 
(1) Ordinary differential equation : A differential equation involving only one independent variable 
and derivatives w.r.t. this single independent variable is called an ordinary differential equation. 
For example : 
????
?? ?? = ?? + ?? ,
?? 4
?? ?? ?? 4
+
?? 2
?? ?? ?? 2
(
????
????
)
2
= ?? ?? 
  
 
 
 
 
 
Types of Differential Equations 
(2) Partial Differential Equation : A differential equation involving partial derivatives w.r.t. more 
than one independent variables is called a partial differential equation. 
For example : 
?
2
?? ??? 2
+
?
2
?? ??? 2
+
?
2
?? ??? 2
= 0 
  
 
 
 
Types of Differential Equations 
(3) Linear and non-linear differential equations : A differential equation is called linear if (i) 
every dependent variable and every derivative involved occurs in the first degree and (ii) no product of 
dependent and/or derivatives occur. A differential equation which is not linear is called non-linear 
differential equations. The general form of a ?? th 
 order linear differential equation is 
?? ?? ( ?? )
?? ?? ?? ?? ?? ?? + ?? ?? -1
( ?? )
?? ?? -1
?? ?? ?? ?? -1
+ ? + ?? 1
( ?? )
????
????
+ ?? 0
( ?? ) ?? = ?? ( ?? ) 
  
 
 
 
Order and Degree of a Differential Equation 
The order of the highest order derivative involved in a differential equation is called the order of the 
differential equation. The degree of a differential equation is the degree of the highest order derivative 
which occurs in it, after the differential equation has been made free of radicals and fractions as far as 
derivatives are concerned.