Download Second-Order Homogeneous Equations There are two

ASSIGNMENT # 02
NAME
CLASS
SECTION
ROLL NO
SUBJECT
QASIM SARWAR
BS (CS)
CS-2
9071
MATHEMATICS
Topic Homogeneous And Non-Homogeneous Equation And
Differentational Equations
Homogeneous Equation
Definition
“An equation that can be rewritten into the form having
zero on one side of the equal sign and a homogeneous function of
all the variables on the other side”.
It is simply an equation where both coefficients of the differentials
dx and dy are homogeneous. To test this you first place the
equation into the differential form:
Differential Equations
“A differential equation is an equation
which contains the derivatives of a variable”, such as the equation
Here x is the variable and the derivatives are with respect to a
second variable t. The letters a, b, c and d are taken to be constants
here. This equation would be described as a second order, linear
differential equation with constant coefficients. It is second order
because of the highest order derivative present, linear because none
of the derivatives are raised to a power, and the multipliers of the
derivatives are constant. If x were the position of an object and t
the time, then the first derivative is the velocity, the second the
acceleration, and this would be an equation describing the motion
of the object. As shown, this is also said to be a non-homogeneous
equation, and in solving physical problems, one must also consider
the homogeneous equation.
First Order Homogeneous Differential Equation
A first order
homogeneous differential equation involves only the first
derivative of a function and the function itself, with constants only
as
multipliers.
The
equation
is
of
the
form
and
can
be
solved
by
the
substitution
The solution which fits a specific physical situation is obtained by
substituting the solution into the equation and evaluating the
various constants by forcing the solution to fit the physical
boundary conditions of the problem at hand. Substituting gives
Differential Equation Terminology
Some general terms used in
the discussion of differential equations
Order
The order of a differential equation is the highest power of
derivative which occurs in the equation, e.g., Newton's second law
produces a 2nd order differential equation because the acceleration
is the second derivative of the position.
Linear and nonlinear
A differential equation is said to be linear if
each term in the equation has only one order of derivative, e.g., no
term has both y and the derivative of y with respect to time. Also,
no derivative is raised to a power.
Homogeneous and non-homogeneous
A differential equation is
said to be homogeneous if there is no isolated constant term in the
equation, e.g., each term in a differential equation for y has y or
some derivative of y in each term.
Homogeneous and non-homogeneous equations
You recall that a linear differential equation
Was called homogeneous if
, and non-homogeneous or
inhomogeneous otherwise. We use the same terminology for
systems of linear equations and for matrix equations:
A matrix equation
Is called homogeneous if is the zero vector (all entries are zero).
A system of linear equations is called homogeneous if the
equivalent matrix equation is homogeneous.
Homogeneous matrix equations have some special properties
1.
The matrix equation
Always has at least one solution, the zero solution
(Here 0 stands for a column vector all of whose entries are zero.)
2.
If the column vectors
matrix equation
and
are two solutions to the
then so is any linear combination of them,
.
3.
The complete solution to a matrix equation
Is always given in the form
Where ,
,.....
are solutions and , ,...., are
parameters. The number of parameters depends on the
dimension of the ``solution space.''
You can see why property (1) holds; a system of linear equations like will
always be satisfied by setting all the variables equal to zero. (This
is the same reason a homogeneous linear differential equation can
always be satisfied by setting)
Property (2) depends on the linearity of multiplication by
. If
Then we have that
Property (3) also really comes from the linearity, since if we have
Then we have that
This is the same reason that the general solution to a homogeneous
linear differential equation is a linear combination of particular
solutions, such as
In the case of differential equations, the number of different
particular solutions, or the number of constants in the general
solution, depends on the order of the differential equation; one
solution for a first order equation, two different solutions for a
second order equation, etc. In the case of matrix equations, the
number of particular solutions is the number of parameters in the
general or complete solution, the dimension of the solution space.
We can also see property (3) in action by solving a matrix equation. Here's
the equation:
The augmented matrix of this equation has the row echelon form
so we can write down the complete general solution
We can rewrite this as
The particular solutions from which we can put together this
complete solution are
The really nice thing we get out of this is a method for finding
solutions to non-homogeneous systems of linear equations (or nonhomogeneous matrix equations.) It works exactly the same way as
solutions for linear differential equations:
If the matrix equation
has one particular solution
equation
, and the associated homogeneous
has the complete solution
, then the complete solution to the
original non-homogeneous equation is
Example
Has the complete solution (which we computed earlier)
Which we can rewrite as
This is the sum of the solution to the associated homogeneous
system, which we wrote down in the previous example,
And a particular solution to this inhomogeneous system
Example
The homogeneous system of linear equations
Has the complete solution
The non-homogenous system
Has one particular solution
To get the complete solution to the non-homogeneous system
We add these together:
First-Order Homogeneous Equations
A function f(x, y) is said to be homogeneous of degree n if the
equation
Holds for all x, y and z (for which both sides are defined)
Example 1
The function f(x, y) = x2 + y2 is homogeneous of
degree 2, since
Example 2
The function
4, since
is homogeneous of degree
Example 3: The function f(x, y) = 2 x + y is homogeneous of
degree 1, since
Example 4
The function f(x, y) = x3 – y2 is not homogeneous,
since
Which does not equal z n f(x, y) for any n.
Example 5
The function f(x, y) = x3 sin ( y/x) is homogeneous of
degree 3, since
A first-order differential equation
is said to be homogeneous if M(x, y) and N(x, y) are both
homogeneous functions of the same degree.
Example 6: The differential equation
Is homogeneous because both M(x, y) = x2 – y2 and N(x, y) = x y
are homogeneous functions of the same degree (namely, 2).
The method for solving homogeneous equations follows from
this fact:

The substitution y = x u (and therefore d y = x d u + u d x)
transforms a homogeneous equation into a separable one.
Example 7
Solve the equation (x2 – y2) d x + x y d y = 0.
This equation is homogeneous, as observed in Example 6. Thus
to solve it, make the substitutions y = x u and d y = x d y + u d x
This final equation is now separable (which was the intention).
Proceeding with the solution,
Therefore, the solution of the separable equation involving x and
v can be written
To give the solution of the original differential equation (which
involved the variables x and y), simply note that
Replacing v by y/ x in the preceding solution gives the final
result:
This is the general solution of the original differential equation.
Example 8: Solve the IVP
Since the functions
Are both homogeneous of degree 1, the differential equation is
homogeneous. The substitutions y = xv and d y = x d v + v d x
transform the equation into
Which simplifies as follows
The equation is now separable. Separating the variables and
integrating gives
The integral of the left-hand side is evaluated after performing
partial fraction decomposition
Therefore,
The right-hand side of (†) immediately integrates to
Therefore, the solution to the separable differential equation is
Now, replacing v by y/ x gives
As the general solution of the given differential equation.
Applying the initial condition y (1) = 0 determines the value of
the constant c
Thus, the particular solution of the IVP is
This can be simplified to
As you can check
Technical note: In the separation step (†), both sides were
divided by (v + 1) ( v + 2), and v = –1 and v = –2 were lost as
solutions. These need not be considered, however, because even
though the equivalent functions y = – x and y = –2 x do indeed
satisfy the given differential equation, they are inconsistent with
the initial condition.
Second-Order Homogeneous Equations
There
are
two
definitions of the term “homogeneous differential equation.” One
definition calls a first-order equation of the form
Homogeneous if M and N are both homogeneous functions of
the same degree. The second definition and the one which you'll
see much more often-states that a differential equation (of any
order) is homogeneous if once all the terms involving the
unknown function are collected together on one side of the
equation, the other side is identically zero. For example,
But
The non-homogeneous equation
Can be turned into a homogeneous one simply by replacing the
right-hand side by 0:
Equation (**) is called the homogeneous equation
corresponding to the non-homogeneous equation, (*). There
is an important connection between the solution of a nonhomogeneous linear equation and the solution of its
corresponding homogeneous equation. The two principal results
of this relationship are as follows:
Theorem A
If y1(x) and y2(x) are linearly independent solutions
of the linear homogeneous equation (**), then every solution is a
linear combination of y1 and y2. That is, the general solution of
the linear homogeneous equation is
Theorem B
If y(x) is any particular solution of the linear nonhomogeneous equation (*), and if y h (x) is the general solution
of the corresponding homogeneous equation, then the general
solution of the linear non-homogeneous equation is
That is,
Note
The general solution of the corresponding homogeneous
equation, which has been denoted here by yh, is sometimes
called the complementary function of the non-homogeneous
equation (*).] Theorem A can be generalized to homogeneous
linear equations of any order, while Theorem B as written holds
true for linear equations of any order. Theorems A and B are
perhaps the most important theoretical facts about linear
differential equations—definitely worth memorizing.
Example 1
The differential equation
Is satisfied by the functions
Verify that any linear combination of y1 and y2 is also a solution
of this equation. What is its general solution?
Every linear combination of y1 = e x and y2 = x ex looks like this:
For some constants c1 and c2. To verify that this satisfies the
differential equation, just substitute. If y = c1 e x + c2 x ex , then
Substituting these expressions into the left-hand side of the
given differential equation gives
Thus, any linear combination of y1 = e x and y2 = xe x does indeed
satisfy the differential equation. Now, since y1 = e x and y2 = xe x
are linearly independent, Theorem A says that the general
solution of the equation is
Example 2: Verify that y = 4 x – 5 satisfies the equation
Then, given that y1 = e− x and y2 = e− 4x are solutions of the
corresponding homogeneous equation, write the general solution
of the given non-homogeneous equation.
First, to verify that y = 4 x – 5 is a particular solution of the nonhomogeneous equation, just substitute. If y = 4 x – 5, then y′ = 4
and y″ = 0, so the left-hand side of the equation becomes
Now, since the functions y1 = e− x and y2 = e− 4x are linearly
independent (because neither is a constant multiple of the other),
Theorem A says that the general solution of the corresponding
homogeneous equation is
Theorem B then says
Example 3 Verify that both y1 = sin x and y2 = cos x
satisfy the homogeneous differential equation y″ + y = 0.
What then is the general solution of the non-homogeneous
equation y″ + y = x?
If y1 = sin x, then y″1 + y1 does indeed equal zero. Similarly, if y2
= cos x, then y″2 = y is also zero, as desired. Since y1 = sin x and
y2 =cos x are linearly independent, Theorem A says that the
general solution of the homogeneous equation y″ + y = 0 is
Now, to solve the given non-homogeneous equation, all that is
needed is any particular solution. By inspection, you can see that
y = x satisfies y″ + y = x. Therefore, according to Theorem B,
the general solution of this non-homogeneous equation is
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