Courses

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

## 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
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
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

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

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
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

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

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
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

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

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
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

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

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.
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
```
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

4 docs

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;