Linear partial differential equations of higher order with constant coefficients
Objective
To study homogeneous linear partial differential equations and methods to solve them.
To study non-homogeneous linear partial differential equations and methods to solve them.
Modules
Module I- Introduction
Module II- The complementary function
Module III- The particular integral of the solution
Module IV- Non-homogeneous linear equations
Introduction
A differential equation involving two or more independent variables and partial derivatives with respect to them is called a partial differential equation. A partial differential equation
in which the partial derivatives occur only in the first degree and not multiplied together is called a linear partial differential equation, otherwise it is said to be non-linear.
A relation between independent variables and the dependent variable (containing arbitrary constants or functions) from which a partial differential equation is formed is called a solution of the partial differential equation.
Linear partial differential equations of higher order with constant coefficients may be divided into two categories as given below.
(i) Equations in which the partial derivatives occurring are all of the same order and
the coefficients are constants. Such equations are called homogeneous linear partial differential equations with constant coefficients.
(ii) Equations in which the partial derivatives occurring are not of the same order and the coefficients are constants. Such equations are called non-homogeneous linear partial differential equations with constant coefficients.
For example,
and
are equations of the first category.
and
are equations of the second category.
The standard form of a homogeneous linear partial differential equation of the nth order with constant coefficients is
(1)
where are constants.
If we use the operators D and for and respectively, we can symbolically write equation (1) as
or(2)
where is a homogeneous polynomial of the nth degree in D and D’.
The method of solving (2) is similar to that of solving ordinary linear differential equations with constant coefficients.
The general solution of (2) is of the form z = complementary function + particular integral, where complementary function (C.F.) is the R.H.S. of the general solution ofand the particular integral (P.I.) is given by .
Module II-
The complementary function
The complementary function of is the R.H.S. of the solution of
(1)
Assume that
(2)
is a solution of (1), where is an arbitrary function.
Differentiating (2) partially with respect to x and then with respect to y, we have
…………..……………
Similarly, and
.
Since (2) is a solution of (1), we have
(3)
Since is arbitrary, .
Therefore (3) reduces to or (4)
Thus will be a solution of (1), if m satisfies the algebraic equation (4) or m is a root of equation (4), which we get by replacing D
by m and D’ by 1 in the equation and dropping z from it.
The equation is called the auxiliary equation, which is an algebraic equation of the nth degree in m and hence will have n roots.
Case 1: The roots are distinct (real or complex).
Let m1, m2, …, mn be the n roots.
The solution of (1) corresponding to these roots are
.
The general solution of (1) is given by a linear combination of these solutions.
i.e., the general solution of (1) is given by
.
Case 2. Two roots are equal and others are distinct.
Let m1, m1, m3, m4, …, mn be the roots.
Here
.
Therefore .
Hence solution of (1) will be a combination of the solutions of the component equations
.
Consider , i.e., , which is a Lagrange’s equation.
The corresponding auxiliary equations are
.
Solving, we get and .
Therefore the general solution of is or .
Now consider(5)
Let(6)
Therefore (5) becomes (7)
The solution of (7) is . Using this value of u in (6), it becomes
or (8)
which is a Lagrange’s equation.
The auxiliary equations are
.
Solving, we get and .
Therefore the solution of (8) and hence (5) is
oror.Therefore the general solution of (1) is
.
Hence the complementary function of the solution of is
.Case 3.r of the roots are equal and others distinct.
Let m1= m2 = m3 = … = mr.
Proceeding as in case 2, we can show that the complementary function of the solution of is
.
Module III-
The particular integral of the solution
As in the case of ordinary differential equations, there are formulas or methods for finding particular integrals (P.I.) of the solution of homogeneous (and also non homogeneous) linear partial differential equations with constant coefficients. The formulas or methods are given below.
Case 1.
P.I.= if
If , then or its power will be a factor of . In this case we factorise and proceed as in ordinary differential equations and use the following results.
,
,…….…….
.
The above results can be derived by using Lagrange’s linear equation method.
For example, let .
i.e., .
The auxiliary equations are
.
The solutions of these equations are and or .
Example. Consider the equation .The symbolic form of this equation is
.
Its auxiliary equation is
or
or.
Therefore and the complementary function is
C.F.= .The particular integral is
P.I.=
= (putting D = 1, D’ = 2)
=.
Therefore the complete solution is
.Case 2. or .
P.I.=
Provided .
If then will be a factor of . In this case we proceed as in ordinary differential equations and use the results
and
.
Example. Consider the equation .
This can be written in symbolic form as .
Its auxiliary equation is
or.
Therefore m = 0, 1 and the complementary function is
C.F.= .
The particular integral is
P.I.=
=
=
=
=
=.
Therefore the complete solution is
.
Case 3. .
P.I. = , where is to be expanded in series of powers of D or D’ and operate on xmyn term by term.
Example. Consider the pde .
This equation may be written as
.
Its auxiliary equation is
or.
The complementary function is
C.F.= .
The particular integral is
P.I.=
=
=
=
=
=
=
=.
Hence the complete solution is
.
Case 4. Q(x,y) is any function of x and y
P.I. = .
Resolve into partial fractions considering as a function of D and operate the following where c is replaced by y + mx after integration.
Example. Consider the equation to solve.
This may be written in symbolic form as
.
Its auxiliary equation is
.
The complementary function is
C.F.= .
The particular integral is
P.I.=
=
=
=
where c is to be replaced by y + mx = y + 2x after integration.
=
=
=
=
where c is to be replaced by y + mx = y - 3x after integration
=
=
=.
Therefore the complete solution is
.
Module IV- Non-homogeneous linear equations
If is not homogeneous in the equation, then it is called non-homogeneous linear pde. Its complete solution = C.F. + P.I.
To find C.F., factorize into the form . Then
.
The auxiliary equations are
.
Therefore,
.
Also,
.
Therefore, .
Hence the C.F. is the addition of the solutions to the various factors.
The methods to find P.I. are the same as those for homogeneous linear equations.
Example. Consider the equation .
Here .
The solution corresponding to the factor is .
Therefore, C.F.= .
The particular integral is
P.I. = .
Hence the complete solution is
.
Summary
In the session we have discussed the methods of finding solution of a homogeneous linear partial differential equation with constant coefficients. The methods are similar to ordinary linear differential equations with constant coefficients.The general solution is of the form z = complementary function + particular integral, where complementary function depends on the nature of roots of the auxiliary equation and the particular integral depends on the right hand side function of the given equation. We discussed three cases of finding the complementary function and four cases of finding particular integrals. Also we have discussed the method of solving a non-homogeneous linear partial differential equation.
Assignment Questions
Solve the following partial differential equations
Quiz
1. The complementary function of is
a.
b.
c.
2. The particular integral of is
a.
b.
c.
3. An example for homogeneous linear partial differential equations with constant coefficients is
a.
b.
c.
Answers
- a2. b3. c
Glossary
Variable: A variable is an object, event, idea, feeling, time period, or any other type of category you are trying to measure.
Independent variable: It is a variable that stands alone and isn't changed by the other variables you are trying to measure.
Dependent variable: It is something that depends on other factors.
Function: It is an assignment f from a set A into another set B; the set A is called domain of f and the set of all function values is called the range of f.
Partial derivative:The ordinary derivative of a function of several variables with respect to one of the independent variables, keeping all other independent variables constant is called the partial derivative of the function with respect to the variable. If z = f(x, y), then if it exists, is said to be the partial derivative of f with respect to x at (a, b) and is denoted by or fx(a, b).
Order: The order of a partial differential equation is the order of the highest partial derivative in the equation.
Degree: The degree of a partial differential equation is the degree of the highest order partial derivative occurring in the equation, after the equation has been made free from radicals and fractions as far as the derivatives are concerned.
FAQs
1. Solve the equation .
Answer.
Here the auxiliary equation is
i.e.,.
Therefore, C.F. = .
P.I.=
=
=
= .
Therefore the general solution of the given equation is
.
2. Solve the equation .
Answer.
Here the auxiliary equation is
Therefore, C.F.=
P.I.=
=
=
=
=
=
= .
Hence the complete solution is
.
3. Solve
Answer.
Here the auxiliary equation is
Therefore, C.F. =
P.I1=
=
=
=
=
P.I2=
=
=
=
= .
Hence the complete solution is
4. Solve
Answer.
Here the auxiliary equation is
.
Therefore, C.F. = .
P.I.=
= 4.
=
=
=
=
=
=
=
Thus the complete solution is
.
5. Solve the equation.
Answer.
The given equation is a non homogeneous linear equation and it may be written as
The corresponding C.F.is
.
P.I.=
=
=
=
=
Therefore, the general solution is
.
6. Solve
Answer.
The given equation is .
The C.F. is
P.I1=
=
=
P.I2=
=
=
=
=
=
= .
Thus the complete solution is
z = C.F. + P.I1 + P.I2.
Reference
- An Elementary Course in Partial Differential Equations, 2nd edn. by T Amaranath, Narosa Publishing House, New Delhi (2003).
- Ordinary and Partial Differential Equations by M.D. Raisinghania,S.Chand, New Delhi(2010).
- Differential Equations and Their Applications, 2nd ed. by Zafar Ahsan, Prentice-Hall, New Delhi (2006).
- Partial Differential Equations and Integral Transforms by T Veerarajan, Tata McGraw-Hill, New Delhi (2004).
- Advanced Engineering Mathematics by E. Kreyszig, John Wylie & Sons, New York (1999).