ENS181 – Engineering Mathematics

UNIT I. INTRODUCTION TO DIFFERENTIAL EQUATIONS

Module 2. Origin and Solutions of Differential

Equations

Engr. Lhemar Jon M. Violango

Instructor
 Overview

Every student of calculus has spent a significant amount of time in finding the
solutions of first-order differential equations of the form

(1)

This antiderivative problem is often written

(2)

and the student is asked to find a single function of x whose derivative is identical
to f(x) on some interval.
 Overview

This important theorem establishes the fact that solutions of equation (1) do not
occur in isolation, but as one-parameter families of solutions, the parameter being the so-
called arbitrary constant c of equation (2).

Having determined such a function it is proved that any other function that satisfies
the differential equation (1) differs from that function by a constant for all x in the
interval.
 Overview

If one considers the general first-order differential equation

(3)

the problem of finding solutions, that is, functions 𝜙(𝑥) that satisfy the equation when
substituted for the dependent variable y, is in general more difficult if not impossible.
However, as we shall see, these solutions, when they exist, occur as one-parameter
families of solutions.
In Unit II we shall study a number of methods for finding families of solutions for
some particular types of first-order equations, but in general there is no method of attack
that will solve every such equation. We content ourselves for the moment by illustrating
what happens in a few simple examples.
 Learning Outcomes

At the end of this module, you must be able to:

✓ Identify the kinds of solutions of a DE.
✓ Verify solutions of differential equations.
✓ Derive differential equations from various origins by elimination of
arbitrary constants.
✓ Plot and interpret solution curves of differential equations.
 Concept and Theories

• Definition

Any function , defined on an interval 𝐼 and possessing at least 𝑛 derivatives that are
continuous on 𝐼 , which when substituted into an 𝑛 th-order ordinary differential
equation reduces the equation to an identity, is said to be a solution of the equation on
the interval.
 Solution of an Ordinary Differential Equation (ODE)

• Interval of Definition

You cannot think solution of an ordinary differential equation without

simultaneously thinking interval. The interval 𝐼 in the definition is variously called the
interval of definition, the interval of existence, the interval of validity, or the domain of
the solution and can be an open interval (𝑎, 𝑏), a closed interval [𝑎, 𝑏], an infinite
interval [𝑎, ∞), and so on.

A solution of a differential equation that is identically zero on an interval 𝐼 is said to

be a trivial solution.
 Solution of an Ordinary Differential Equation (ODE)

A function 𝑦 = 𝑔(𝑥) is a solution of a given differential equation on some

interval, 𝑎 < 𝑥 < 𝑏, if it is defined and differentiable throughout the interval and is
such that the equation becomes an identity, when 𝑦 and 𝑦’ are replaced by 𝑔 and 𝑔’
respectively.

Illustrative Example: Determine whether 𝑦 = 3𝑒 −𝑥 is a solution of 𝑦 ′ + 𝑦 = 0.

Therefore, 𝑦 = 3𝑒 −𝑥 is a solution since it

satisfies the given d.e. 𝑦 ′ + 𝑦 = 0.
 Families of Solution

Types of Solution:

• General solution – a solution containing a number of arbitrary constants equal to the

order of the equation.

• Particular solution – each individual solution of a differential equation (the constants

have values)

• Complete solution – set of all solutions of the given equation.

 Families of Solution

Example 1.2.1
The differential equation (4)

has the family of solutions (5)

simple antiderivative having produced this result. General solution

If we wish to find one member of the family (5) that satisfies the additional condition that 𝑦 = 6 when
𝑥 = 0, we are forced to choose 𝑐 = 8. We then say that

is the solution of the initial value problem Particular solution

 Families of Solution

Example 1.2.2
Consider the second-order equation (6)

Integration of both sides of this equation with respect to 𝑥 yields (7)

A second integration produces General solution (8)

In this example there are two arbitrary constants, so we have a two-parameter family of solutions. This
means that to single out one member of this family we need to provide two pieces of information. These are
usually given by specifying the values of both 𝑦 and 𝑦’ for the same value of 𝑥.

For example, suppose we want the solution to (6) that also satisfies 𝑦(0) = 1, and 𝑦’(0) = 2.
Substituting 𝑥 = 0, and 𝑦’ = 2 into (7) we see that 𝑐1 = 2, so that

Finally, substituting 𝑥 = 0, 𝑦 = 1, we see that 𝑐2 = 1 so that the required solution is

Particular solution
 Families of Solution

Example 1.2.3
𝑑3𝑦 𝑑𝑦
Determine whether 𝑦 = 𝑒 −2𝑥 is a solution of −3 + 2𝑦 = 0.
𝑑𝑥 3 𝑑𝑥

Since it results to an identity,

therefore it is a solution.
 Families of Solution

Example 1.2.4
Determine whether 𝑦 = 2𝑥 2 is a solution of 𝑦 ′ + 𝑦 2 = 0.

Since it does not result to an identity,

therefore it is a NOT solution.
 Families of Solution

Example 1.2.5
Determine whether 𝑦 = 2 cos 2𝑥 is a solution of 𝑦 ′′ + 4𝑦 = 0.

Since it results to an identity,

therefore it is a solution.
 Geometric Interpretation

We saw that a first-order equation usually has a family of solutions. A useful

technique in understanding the nature of these solutions is to graph representative
solutions from this family.

The graph of a solution of an ODE is called a solution curve. Since is a differentiable

function, it is continuous on its interval 𝐼 of definition. Thus there may be a difference
between the graph of the function and the graph of the solution . Put another way, the
domain of the function need not be the same as the interval 𝐼 of definition (or domain)
of the solution. The example on the next slide illustrates the difference.
 Geometric Interpretation

Function vs Solution
 Geometric Interpretation

Illustrative Example
Graph several members of the family of solutions of the equation

Integrating results to the family solution

Graphing the solutions corresponding to

𝑐 = 2, 1, 0, −1 we obtain the figure on the right.

It is not difficult to imagine what the rest of

the family looks like.
 Explicit and Implicit Solutions

You should be familiar with the terms explicit functions and implicit functions from
your study of calculus. A solution in which the dependent variable is expressed solely in
terms of the independent variable and constants is said to be an explicit solution.

For our purposes, let us think of an explicit solution as an explicit formula 𝑦 = 𝜙(𝑥)
that we can manipulate, evaluate, and differentiate using the standard rules. We can
1 4 𝑥 1
verify that , 𝑦 = 𝑥 , 𝑦 = 𝑥𝑒 , and 𝑦 = are, in turn, explicit solutions of
16 𝑥
 Explicit and Implicit Solutions

Moreover, the trivial solution 𝑦 = 0 is an explicit solution of all three equations.

When we get down to the business of actually solving some ordinary differential
equations, you will see that methods of solution do not always lead directly to an explicit
solution 𝑦 = 𝜙(𝑥) . This is particularly true when we attempt to solve nonlinear first-
order differential equations. Often we have to be content with a relation or expression
𝐺 𝑥, 𝑦 = 0 that defines a solution 𝜙 implicitly.

A relation 𝐺 𝑥, 𝑦 = 0 is said to be an implicit solution of an ordinary differential

equation (3) on an interval 𝐼, provided that there exists at least one function 𝜙 that
satisfies the relation as well as the differential equation on 𝐼.
 Origin of Differential Equations

▪ Geometric Problems
Example 1.2.6
Derive a differential equation from
the orthogonal trajectories of the family
of curves of whether 𝑦 = 𝑐𝑥 2 , where 𝑐 is
constant.

when 𝑥 ≠ 0
 Origin of Differential Equations

▪ Geometric Problems
Example 1.2.7
Derive a differential equation from
the orthogonal trajectories of the family
𝑐
of curves of whether 𝑦 = , where 𝑐 is
𝑥
constant.

when 𝑥 ≠ 0
 Origin of Differential Equations

▪ Geometric Problems
Example 1.2.8
Derive a differential equation from
the family of circles

(𝑥 − 2)2 +(𝑦 + 1)2 = 𝑐 2 ,

where 𝑐 is constant.

when y ≠ −1
 Origin of Differential Equations

▪ Physical Problems

A differential equation can be obtained as mathematical models of physical problems.

Examples,

✓ Population Dynamics (Growth and Decay problems)

✓ Newton’s Law of Cooling and Heating
✓ Motion problems
✓ Mixing problems
✓ Electrical circuit problems, etc.
 Origin of Differential Equations

▪ Primitive Origin
Differential equations can be obtained from a relation of variables involving 𝑛 essential arbitrary
constants, where 𝑛 also stands for the number of times the derivatives of a dependent variable be taken
which is free of arbitrary constants. This means, the arbitrary constants has to be eliminated.

Example 1.2.9
Obtain the differential equation associated with
the primitive
𝑦 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶

where 𝐴, 𝐵 and 𝐶 are constants.

 Origin of Differential Equations

▪ Primitive Origin
Example 1.2.10
Obtain the differential equation associated with
the primitive
𝑥 3 − 3𝑥 2 𝑦 = 𝑐

where 𝑐 is a constant.
 Origin of Differential Equations

▪ Primitive Origin
Example 1.2.11
Obtain the differential equation associated with
the primitive
𝑎
𝑟=
sin 𝜃+cos 𝜃

where 𝑎 is a constant.
 Origin of Differential Equations

▪ Primitive Origin
Example 1.2.12
Obtain the differential equation associated
with the primitive
𝑥2𝑦3 + 𝑥3𝑦5 = 𝑐

where 𝑐 is a constant.
 References

1. Zill, Dennis G. A First Course in Differential Equations with Modeling Applications.

10th edition. 2012. Brooks/Cole Cengage Learning, USA.
2. Kreyzig, Erwin. Advanced Engineering Mathematics. 10th edition. 2011. John Wiley
& Sons, Inc.