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EDUCATION HOLE PRESENTS
Engineering Mathematics – II
Unit-I

Differential Equations.............................................................................................................. 2
Linear differential equations of nth order with constant coefficients..................................................................2
Complementary function and Particular integral..............................................................................4
To find the Complementary Function...................................................................................................................4
Particular Integral.............................................................................................................................................4
Simultaneous linear differential equations.......................................................................................5
Using the D Operator............................................................................................................................................5
Using Laplace Transform.......................................................................................................................................6
Solution of second order differential equations by changing dependent & independent variables....................7
Normal form of differential equation ...............................................................................................8
Types of Differential Equations.............................................................................................................................9
Methods................................................................................................................................................................9
Ordinary Differential Equations........................................................................................................................9
Linear Differential Equation Solutions............................................................................................................10
First Order Linear Equations...........................................................................................................................10
Method of variation of parameters....................................................................................................................10
Applications to engineering problems ............................................................................................ 13
Differential Equations
Linear differential equations of nth order with constant coefficients
The linear homogeneous differential equation of the n-th order with constant coefficients can be
written as
where a1, a2,..., an constants which may be real or complex.
Using the linear differential operator L(D), this equation can be represented as
where
For each differential operator with constant coefficients, we can introduce the characteristic
polynomial

The algebraic equation
is called the characteristic equation of the differential equation.
According to the fundamental theorem of algebra, a polynomial of degree n has exactly n roots,
counting multiplicity. In this case the roots can be both real and complex (even if all the
coefficients of a1, a2,..., an are real).
Let us consider in more detail the different cases of the roots of the characteristic equation and
the corresponding formulas for the general solution of differential equations.
Case 1. All Roots of the Characteristic Equation are Real and Distinct
Assume that the characteristic equation L(λ) = 0 has n roots λ1, λ2,..., λn. In this case the general
solution of the differential equation is written in a simple form:
where C1, C2,..., Cn are constants depending on initial conditions.
Case 2. The Roots of the Characteristic Equation are Real and Multiple
Let the characteristic equation L(λ) = 0 of degree n have m roots λ1, λ2,..., λm, the multiplicity of
which, respectively, is equal to k1, k2,..., km. It is clear that the following condition is satisfied:
Then the general solution of the homogeneous differential equations with constant coefficients
has the form
It is seen that the formula of the general solution has exactly ki terms corresponding to each root
λi of multiplicity ki. These terms are formed by multiplying x to a certain degree by the
exponential function exp(λi x). The degree of x varies in the range from 0 to ki − 1, where ki is
the multiplicity of the root λi.
Case 3. The Roots of the Characteristic Equation are Complex and Distinct
If the coefficients of the differential equation are real numbers, the complex roots of the
characteristic equation will be presented in the form of conjugate pairs of complex numbers:
In this case the general solution is written as

Case 4. The Roots of the Characteristic Equation are Complex and Multiple
Here, each pair of complex conjugate roots α ± iβ of multiplicity k produces 2k particular
solutions:
Then the part of the general solution of the differential equation corresponding to a given pair of
complex conjugate roots is constructed as follows:
Complementary function and Particular integral
Standard Form
where are constants.
The general solution of such a differential equation contains two parts; the Complementary
Function and the Particular Integral.
To find the Complementary Function
Write the auxiliary equation in the form: where the differential equation
is in the standard form.
Solving the auxiliary equation, a quadratic, will yield two roots, say m1 and m2.
The Complementary Function (CF) is then written based on the roots.
• If m1 and m2 are real and different, then the CF is
• If m1 and m2 are real and equal, ie m1 = m2 = m, then the CF is
• If m1 and m2 are complex roots of the form and , then the CF is
where and are arbitrary constants.
Particular Integral
The Particular Integral is similar to the function of x, on the RHS of the differential
equation and it is determined by the method of undetermined coefficients.

If the RHS of the equation is a constant, then we use a trial PI of K (a different constant). A trial
PI of a general first degree form and a general second degree form
is used for functions of and respectively. If the function is of the form ,
then we use , the same exponential function. Similarly, trigonometric functions will have
trial PIs of the form .
are undetermined coefficients. By finding and and substituting and
into the given differential equation, the values of the undetermined coefficients can be
obtained. Hence the Particular Integral is found.
Combine the Complementary Function and the Particular Integral and we will get the General
Solution of the Second Order Differential Equation with constant coefficients
Simultaneous linear differential equations
A simultaneous differential equation is one of the mathematical equations for an indefinite
function of one or more than one variables that relate the values of the function. Differentiation
equation in various orders, Differential equations plays an important function in engineering,
physics, economics, and other disciplines. This analysis concentrates on linear equations with
Constant Coefficients.
Using the D Operator
The D operator is a linear operator applied to functions and which is defined as .
For example
Example:
Example - First example
Problem
Workings
The equations may be written as:

(1)
(2)
Eliminating y from equations (1) and (2)
i.e.
Solution
From equation (1)
Therefore
Using Laplace Transform
Laplace transform or the Laplace operator is a linear operator applied to functions and which is
defined as where
Problem
Apply the Laplace Transform and find and :
Give that at
Workings
The equations may be written as:
Hence
Therefore

(3)
(4)
Then we multiply the equation (3) with :
(5)
We subtract equation (4) from equation (5) :
Solution
Therefore
From equation (4) we get :
Solution of second order differential equations by changing dependent &
independent variables
We suppose that we have a second order differential equation with dependent variable u and independent
variable t. The harmonic oscillator or pendulums are good examples. We suppose, further, that we are
given initial values of u and u', as well as a formula for u" in terms of u, u' and t
u" = f(u, u', t)
If f does not involve u or u' we could integrate both sides of the equation once to find u', and again to find
u, with a linear function ct + d that must be determined from the initial values
Thus the problem we face is not unlike that of performing a double integral. However, it is surprisingly
easy to do, on a spreadsheet.
We will use an approximation technique that has the order of accuracy of the trapezoid rule;
improvements can be obtained by extrapolating as with most numerical methods.
We begin by describing the basic approach.
We will use a column to represent each the variables t, u, u' and u". The first three of these will start at the
given initial values of these variables; and the initial value of u" can be computed from them.

In each successive row, t will increase by a constant amount, d. (You can later choose to make d smaller
than your initial choice when variables change too much in one d interval.)
We will set
(Which are generic statements, independent of the equation itself.)
Finally we provide an expression for u"(t + d). In doing so we cannot use u(t + d) or u'(t + d) or else our
definitions would be circular. On the other hand we would like to use something which averages the
second derivative at the ends of the time interval between t and t + d (at least to some order), just as the is
done above for u and u'. For the value of u" at t we use f(t, u(t), u'(t)). For the value of u" at t + d we use
u"(t + d) = f(t + d, u(t) + d * u'(t), u'(t) + d * f(t, u(t), u'(t)))
Thus we use the linear approximation to u and u' in f(t + d, u, u') defined at t and evaluated at t + d. We
could do slightly better by using a quadratic approximation
Which would be correct to second order, but the trapezoid rule is already in error in second order, so this
will not do much good, usually.
You can observe the behavior of u and u' as a function of t by plotting the first three columns (using the
"charting" capability of the spreadsheet program with xy scatter plots). You can observe the "phase plane"
behavior of the solution by plotting the second and third columns, namely u' vs. u.
If you do things right you can change d or parameters in f in one keystroke, and watch how the solutions
change as you change them.
You can handle equations with several dependent variables, like those of planetary motion, by the same
approach; with motion in the xy plane you can have a column for t, x, y, x', y', x" and y", and can observe
trajectories and look at behavior in the xy plane as you change parameters.
Normal form of differential equation
A differential equation is an equation that relates a function to one or more of its derivatives. Differential
equations are especially applicable when the tools of algebra, which are ideally suited for static systems,
are not enough. Many physical systems are modeled by solving differential equations, although their
usefulness extends well into other fields of science such as chemistry and economics.

Types of Differential Equations
There are two main types of differential equations: Ordinary Differential Equations and Partial
Differential Equations.[1]
The former is simpler of the two, as it can be written in the normal form
for a simple function y = g(x)
The function F consists of the function y and its derivatives up to the nth order. Notice that y is comprised
of only one independent variable x. A differential equation is considered ordinary if the function y in F is
dependent on only one variable. It is important to note that, while most ordinary differential equations can
be written in the normal form (isolating the highest derivative on one side of the equation and moving all
other variables to the other), there are equations in which this cannot be done.
If y were a function of multiple variables, for example
y = F(u,v)
then the derivatives of y in the ordinary equation may be partial derivatives with respect to either u or v. In that
case, any differential equation that has partial derivatives is called a partial differential equation. For
example, the 1-dimensional wave equation[2]
:
Obviously, partial differential equations are much more complicated to solve.
Methods
There are many ways to find solutions to differential equations.
Ordinary Differential Equations
The simplest differential equations to solve are separable differential equations. A differential equation is
separable if it can be written in the form
Then we can separate the two variables, collect the x's on one side and the y's on the other side, then
integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found
the function y. For example, for the derivative n = 1:

The solution is then given implicitly by the expression:
where C is an arbitrary constant.
Linear Differential Equation Solutions
If a differential equation can be written in the normal form
It is considered a linear differential equation.
First Order Linear Equations
The order of a differential equation is equal to the degree of the highest derivative in the equation. For
example, the above equations are order n equations. A first order linear equation appears in the form:
Method of variation of parameters
This method has no prior conditions to be satisfied. Therefore, it may sound more general than
the previous method. We will see that this method depends on integration while the previous one
is purely algebraic which, for some at least, is an advantage.
Consider the equation

In order to use the method of variation of parameters we need to know that is a set of
fundamental solutions of the associated homogeneous equation y'' + p(x)y' + q(x)y = 0. We know
that, in this case, the general solution of the associated homogeneous equation is
. The idea behind the method of variation of parameters is to look for a
particular solution such as
where and are functions. From this, the method got its name.
The functions and are solutions to the system
,
which implies
,
where is the wronskian of and . Therefore, we have
Summary:Let us summarize the steps to follow in applying this method:
(1)
Find a set of fundamental solutions of the associated homogeneous equation
y'' + p(x)y' + q(x)y = 0.
;
(2)
Write down the form of the particular solution
;

(3)
Write down the system
;
(4)
Solve it. That is, find and ;
(5)
Plug and into the equation giving the particular solution.
Example: Find the particular solution to
Solution: Let us follow the steps:
(1)
A set of fundamental solutions of the equation y'' + y = 0 is ;
(2)
The particular solution is given as
(3)
We have the system
;
(4)
We solve for and , and get
Using techniques of integration, we get

;
(5)
The particular solution is:
,
or
Remark: Note that since the equation is linear, we may still split if necessary. For example, we
may split the equation
,
into the two equations
then, find the particular solutions for (1) and for (2), to generate a particular solution for the
original equation by
Applications to engineering problems
• Uniquely covers mathematical modeling, the numerical solution, ordinary and partial
differential equations, and both finite difference and finite element methods
• Introduces readers to scientific computing, which is the integrated science of solving
problems with mathematical, numerical, and programming tools
• Originally written for the course 'Numerical Solution of Differential Equations' offered
at KTH, Stockholm and has been classroom tested over several years
• Features a 'Five M' approach: Modeling, Mathematics, Methods, MATLAB®, and Multi
physics.

• Supplemented with a related Web Site that features solutions to problems, demos,
MATLAB programs, etc.

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