Lecture 1 - Existence and Uniqueness of Solutions of first order differential equations Engineering Mathematics Notes | EduRev

Ordinary Differential Equations- Order, Degree, Formation

Engineering Mathematics : Lecture 1 - Existence and Uniqueness of Solutions of first order differential equations Engineering Mathematics Notes | EduRev

 Page 1


Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 1 
 
 
 
 
 
 
 
 
 
Paper: Ordinary Differential Equations 
Lesson: Existence and Uniqueness of Solutions of first 
order differential equations 
Lesson Developer: Dr. Sada Nand Prasad 
College/Department: Department of Mathematics, A.N.D. 
College, University of Delhi 
 
 
 
 
 
 
 
 
Page 2


Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 1 
 
 
 
 
 
 
 
 
 
Paper: Ordinary Differential Equations 
Lesson: Existence and Uniqueness of Solutions of first 
order differential equations 
Lesson Developer: Dr. Sada Nand Prasad 
College/Department: Department of Mathematics, A.N.D. 
College, University of Delhi 
 
 
 
 
 
 
 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 2 
 
Table of Contents: 
Chapter: Existence and Uniqueness of Solutions of first order differential equations 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Equations which can be Factorized or solvable for p 
? 4. Equations which cannot be Factorized 
                   4.1. Equations Solvable for x 
                   4.2. Equations Solvable for y 
? 5. Clairaut's Equation 
? 6. Singular Solution 
? 7. Envelops and Orthogonal trajectories 
? 8. Existence and Uniqueness of Solutions of first order    
        differential equations 
? Summary 
? Exercises 
? References/ Further Reading 
 
Page 3


Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 1 
 
 
 
 
 
 
 
 
 
Paper: Ordinary Differential Equations 
Lesson: Existence and Uniqueness of Solutions of first 
order differential equations 
Lesson Developer: Dr. Sada Nand Prasad 
College/Department: Department of Mathematics, A.N.D. 
College, University of Delhi 
 
 
 
 
 
 
 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 2 
 
Table of Contents: 
Chapter: Existence and Uniqueness of Solutions of first order differential equations 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Equations which can be Factorized or solvable for p 
? 4. Equations which cannot be Factorized 
                   4.1. Equations Solvable for x 
                   4.2. Equations Solvable for y 
? 5. Clairaut's Equation 
? 6. Singular Solution 
? 7. Envelops and Orthogonal trajectories 
? 8. Existence and Uniqueness of Solutions of first order    
        differential equations 
? Summary 
? Exercises 
? References/ Further Reading 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 3 
 
1. Learning Outcomes: 
After completing this chapter, we can 
? resolve the differential equations into rational and solve it. 
? solve equations for p, x and y. 
? explain Clairaut's equation and find its solution 
? find the singular solution of a differential equation 
? Find the envelopes and orthogonal trajectories of the family of different surfaces 
? state the existence and uniqueness theorem for first order ordinary differential 
equations and use it to know whether the solution exist and unique or not. 
Page 4


Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 1 
 
 
 
 
 
 
 
 
 
Paper: Ordinary Differential Equations 
Lesson: Existence and Uniqueness of Solutions of first 
order differential equations 
Lesson Developer: Dr. Sada Nand Prasad 
College/Department: Department of Mathematics, A.N.D. 
College, University of Delhi 
 
 
 
 
 
 
 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 2 
 
Table of Contents: 
Chapter: Existence and Uniqueness of Solutions of first order differential equations 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Equations which can be Factorized or solvable for p 
? 4. Equations which cannot be Factorized 
                   4.1. Equations Solvable for x 
                   4.2. Equations Solvable for y 
? 5. Clairaut's Equation 
? 6. Singular Solution 
? 7. Envelops and Orthogonal trajectories 
? 8. Existence and Uniqueness of Solutions of first order    
        differential equations 
? Summary 
? Exercises 
? References/ Further Reading 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 3 
 
1. Learning Outcomes: 
After completing this chapter, we can 
? resolve the differential equations into rational and solve it. 
? solve equations for p, x and y. 
? explain Clairaut's equation and find its solution 
? find the singular solution of a differential equation 
? Find the envelopes and orthogonal trajectories of the family of different surfaces 
? state the existence and uniqueness theorem for first order ordinary differential 
equations and use it to know whether the solution exist and unique or not. 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 4 
 
2. Introduction: 
We have used many methods to solve differential equations of first order and of degree 
1, e.g., differential equations that can be separated in different variables, exact 
differential equations, equations that can be reduced to homogeneous equation and 
those equations that become exact when we multiply them by some integrating factor. 
In this chapter, we will keep our discussion on the differential equations which are of 
first order but of higher degree. 
Let us take 
dy
p
dx
? , then the general form of the first order and nth degree differential 
equation is given by the equation  
 
12
12
... 0 (1)
n n n
n
p A p A p A
??
? ? ? ? ?  
where 
1
A , 
2
A ,...,
n
A are functions of x and y. 
It is not simple to find the solution of eqn. (1) in its general form. In this chapter we 
consider only that type of eqn. (1) which we can solve easily and explain the methods 
for solving those equations. 
Also we shall discuss the Clairaut equation, the singular solution of the differential 
equations, the envelopes and orthogonal trajectories of the family of surfaces and state 
and the existence and uniqueness theorem for first order ordinary differential equations 
in this chapter. 
Page 5


Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 1 
 
 
 
 
 
 
 
 
 
Paper: Ordinary Differential Equations 
Lesson: Existence and Uniqueness of Solutions of first 
order differential equations 
Lesson Developer: Dr. Sada Nand Prasad 
College/Department: Department of Mathematics, A.N.D. 
College, University of Delhi 
 
 
 
 
 
 
 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 2 
 
Table of Contents: 
Chapter: Existence and Uniqueness of Solutions of first order differential equations 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Equations which can be Factorized or solvable for p 
? 4. Equations which cannot be Factorized 
                   4.1. Equations Solvable for x 
                   4.2. Equations Solvable for y 
? 5. Clairaut's Equation 
? 6. Singular Solution 
? 7. Envelops and Orthogonal trajectories 
? 8. Existence and Uniqueness of Solutions of first order    
        differential equations 
? Summary 
? Exercises 
? References/ Further Reading 
 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 3 
 
1. Learning Outcomes: 
After completing this chapter, we can 
? resolve the differential equations into rational and solve it. 
? solve equations for p, x and y. 
? explain Clairaut's equation and find its solution 
? find the singular solution of a differential equation 
? Find the envelopes and orthogonal trajectories of the family of different surfaces 
? state the existence and uniqueness theorem for first order ordinary differential 
equations and use it to know whether the solution exist and unique or not. 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 4 
 
2. Introduction: 
We have used many methods to solve differential equations of first order and of degree 
1, e.g., differential equations that can be separated in different variables, exact 
differential equations, equations that can be reduced to homogeneous equation and 
those equations that become exact when we multiply them by some integrating factor. 
In this chapter, we will keep our discussion on the differential equations which are of 
first order but of higher degree. 
Let us take 
dy
p
dx
? , then the general form of the first order and nth degree differential 
equation is given by the equation  
 
12
12
... 0 (1)
n n n
n
p A p A p A
??
? ? ? ? ?  
where 
1
A , 
2
A ,...,
n
A are functions of x and y. 
It is not simple to find the solution of eqn. (1) in its general form. In this chapter we 
consider only that type of eqn. (1) which we can solve easily and explain the methods 
for solving those equations. 
Also we shall discuss the Clairaut equation, the singular solution of the differential 
equations, the envelopes and orthogonal trajectories of the family of surfaces and state 
and the existence and uniqueness theorem for first order ordinary differential equations 
in this chapter. 
Existence and Uniqueness of Solutions of first order differential equations 
 
Institute of Lifelong Learning, University of Delhi                                                Pg. 5 
 
3. Equations which can be factorized:  
The general form of the first order and nth degree differential equation is given by, 
 
12
12
... 0
n n n
n
p A p A p A
??
? ? ? ? ? ? 
where 
1
A , 
2
A ,...,
n
A are functions of the variables x and y. 
Now there are two possibilities: 
(a) If we resolve 
12
12
...
n n n
n
p A p A p A
??
? ? ? ? into rational factors of degree 1, then it 
can be written as 
 ? ? ? ? ? ?
12
... 0 (2)
n
p f p f p f ? ? ? ? 
where 
1
f , 
2
f ,...,
n
f are functions of the variables x and y. 
Since all those values of y, for which the factors in eqn. (2) become zero, will satisfy 
eqn. (1). Hence, to solve eqn. (1), we will have to equate each of the factors given in 
eqn. (2) to zero. i.e., 
 0, 1,2,..., (3)
r
p f r n ? ? ? 
If 
 ? ? , , 0, 1,2,..., (4)
rr
F x y C r n ?? 
where , 1,2,..., constants.
r
C r n arearbitrary ? 
are the solutions for eqns. (3), then the general solution of eqn. (1) is given by 
 ? ? ? ? ? ?
1 1 2 2
, , . , , ... , , 0, (5)
nn
F x y C F x y C F x y C ? 
All the constants in eqn. (5), namely, 
12
, ,...,
n
C C C can have infinite number of values, 
so all these solutions given by eqn. (5), will remain general even if we take 
12
...
n
C C C C ? ? ? ? . Hence, the general solution is given by 
 ? ? ? ? ? ?
12
, , . , , ... , , 0, (6)
n
F x y C F x y C F x y C ? 
(b) When the left-hand side of eqn. (1) cannot be factorized. We will take this possibility 
in the next section. 
Value Additions: 
Since we are dealing with the first order differential equation, the general solution should 
contain only one arbitrary constant. There is no loss of generality by replacing the n 
arbitrary constants by a single arbitrary constant.  
 
Example 1: Solve the differential equation 
2 2 2
6 0,
dy
x p xy p y wherep
dx
? ? ? ? . 
Solutions: We have  
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