Flashcards: Differential Equations

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

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``` 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.

```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Flashcards: Differential Equations - Mathematics (Maths) for JEE Main & Advanced

 1. What are the basic types of differential equations?
Ans. The basic types of differential equations are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable, while PDEs involve functions of multiple variables.
 2. How are differential equations used in real-life applications?
Ans. Differential equations are used in various real-life applications such as physics, engineering, biology, and economics to model and analyze dynamic systems and processes.
 3. What is the order of a differential equation?
Ans. The order of a differential equation is the highest order derivative present in the equation. For example, in the differential equation dy/dx + 2y = 0, the order is 1.
 4. How can one solve a first-order differential equation?
Ans. First-order differential equations can be solved using methods like separation of variables, integrating factors, or by using specific formulas depending on the type of the equation.
 5. What is the role of initial conditions in solving differential equations?
Ans. Initial conditions are necessary to find the particular solution of a differential equation. These conditions provide specific values at a given point, helping determine the constants of integration and yielding a unique solution.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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