Unit 4 | unit 4 systems of linear differential equations

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PDE

Unit - 4

Systems of linear differential equations

4.1 Systems of linear differential equations, Types of linear systems

The general linear system of two first order differential equations in two unknown functions x and y is of the form

For example:

Note- When a set of linear equations which is related to a group of functions to their derivative, then the system is called the system of linear differential equations.

The general linear system of three first-order differential equations in three unknown functions x, y and z can be defined as

Example:

The normal form in the case of two linear differential equations in two unknown functions are given as-

For example:

The normal form in the case of linear system of three differential equations in three unknown functions x,y and z are

For example:

Normal form of LDE-

The normal form of in the case of two LDE in two unknown functions is-

The normal form of in the case of a linear system of three differential equations in three unknown functions x, y and z is-

Matrix form of system of linear differential equations

Suppose the system of linear differential equations is gives as below

This system is written as

Where

And

Key takeaways

When a set of linear equations which is related to a group of functions to their derivative, then the system is called the system of linear differential equations.
The normal form in the case of two linear differential equations in two unknown functions are given as-

4.2 Differential operators, an operator method for linear systems with constant coefficients

In order to solve a linear system with constant coefficient, we use symbolic operators.

Suppose x is n-times differentiable function of the independent variable t. We denote the operation of differentiation with respect to t by the symbol ‘D’ and we call it as differential operator.

In terms of this differential operator the derivative dx/dt is denoted by Dx.

Which is

In same order we denote the second derivative of x with respect to t by

The nth derivative of x with respect to t by

That is

Extending the operator notations, we write

And

Where a, b and c are the constants.

In this notation the linear differential expression with constant coefficients

Can be written as-

Here note that the operator in this expression do not represent quantities that are to be multiplied by the function x, but rather they indicate operations that are to be carried out upon this function.

The expression

Where are constants, is called linear differential operator with constant coefficients.

Where are constants, now suppose are both n-times differentiable functions of t and are constants.

Then it can be written as

For example, if the operator is applied to then

An operator method for linear system with constant coefficient

We consider a linear system of the form

Where are linear differential operators with constant coefficient.

Where a, b, are constants.

Example:

A simple example of a system which may be expressed in the form of (1) and (2) is provided by

Let’s denote the above system in notation format

Where

Now we apply the operator to (1) and to (2), we get

Now subtract the second of these equations from first, since

We get

is the left side of this equation is itself a linear differential operator with constant coefficients.

We assume that it is neither a zero nor a non-zero constant and denote it by .

And if we again assume that is the right side and are the members of this, then this member is some function let say .

Then the equation (3) becomes

Equation (4) is a linear differential equation with constant coefficient in the single dependent variable x.

Hence we notice that the procedure has eliminated the other dependent variable y.

We now solve the differential equation (4) for x.

Suppose (4) is of order n. Then the general solution of (4) will be

Where are the n linearly dependent solutions of the homogenous linear equation and are arbitrary constants and is the particular solution (PI).

Now again we apply the operators and to the equations (1) and (2) respectively.

The system becomes

Subtracting, we get

As same as above

The general solution

Where are the n linearly dependent solutions of the homogenous linear equation and are arbitrary constants and is the particular solution (PI).

Key takeaways

In terms of this differential operator the derivative dx/dt is denoted by Dx.

Which is

Can be written as-

4.3 Basic Theory of linear systems in normal form

Let us consider a basic type of system of two linear differential equations in two unknown functions. This is of the form

We will assume that the functions are all continuous on a real interval

If and are zero for all t, then the above system is called homogeneous, otherwise the system is called non-homogeneous.

For example:

Is homogenous,

And the system

Is non-homogeneous.

We mean by the solution of equation (1), an ordered pair of real functions each having a continuous derivative on the real interval such that

For all t such that ,

In other words,

Simultaneously satisfy both equation of the system (1) identically for

Key takeaways

If and are zero for all t, then the above system is called homogeneous, otherwise the system is called non-homogeneous.

4.4 Homogeneous linear systems with constant coefficients: Two Equations in two unknown functions

Linear differential equation are those in which the independent variable and its derivative occur only in the first degree and are not multiplied together.

Thus the general linear differential equation of the n’th order is of the form

Where and X are function of x.

Linear differential equation with constant co-efficient are of the form-

Where are constants.

Rules to find the complementary function-

To solve the equation-

This can be written as in symbolic form-

Or-

It is called the auxiliary equation.

Let be the roots-

Case-1: If all the roots are real and distinct, then equation (2) becomes,

Now this equation will be satisfied by the solution of

This is a Leibnitz’s linear and I.F. =

Its solution is-

The complete solution will be-

Case-2: If two roots are equal

Then complete solution is given by-

Case-3: If one pair of roots be imaginary, i.e. then the complete solution is-

Where and

Case-4: If two points of imaginary roots be equal-

Then the complete solution is-

Example-Solve

Sol.

Its auxiliary equation is-

Where-

Therefore the complete solution is-

Inverse operator-

is that function of x, not containing arbitrary constants which when operated upon gives X.

So that satisfies the equation f(D)y = X and is, therefore, its P.I.

f(D) and 1/f(D) are inverse operator.

Note-

Rules for finding the particular integral-

Let us consider the equation-

Or in symbolic form-

So that-

Now-

Case-1: When X =

In case f(a) = 0, then we see that the above rule will not work,

So that-

Example: Find the P.I. Of (D + 2)

Sol.

P.I. =

Now we will evaluate each term separately-

And

Therefore-

Example: Solve (D – D’ – 2 ) (D – D’ – 3) z =

Sol.

The C.F. Will be given by-

Particular integral-

Therefore the complete solution is-

Case-2: when X = sin(ax + b) or cos (ax + b)

In case then the above rule fails.

Now-

And if

Similarly-

Example: Find the P.I. Of

Sol.

Example: Find the P.I. Of (D + 1) (D + D’ – 1)z = sin (x + 2y)

Sol.

Example: Find P.I. Of

Sol. P.I =

Replace D by D+1

Put

Problem:

Solve

Solution:

Auxiliary equation

Complementary function

Complete Solution is

Problem:

Solve

Solution:

Auxiliary equation

C.F is

[]

The Complete Solution is

Example:

Solve

Solution:

The Auxiliary equation is

The C.F is

P.I

The Complete Solution is

Example:

Solve

Solution:

The Auxiliary equation is

The C.F is

P.I

Now,

The Complete Solution is

Example:

Solve

Solution:

The auxiliary solution is

The C.F is

Now,

[Putting ]

Similarly, we find that

The Complete Solution is

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

[Put ]

The Complete Solution is

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

But

The Complete Solution is

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

[By parts]

The Complete Solution is

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

Here , , . Let

Now,

Put

Multiply by in the numerator and denominator

Put

The Complete Solution is

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

Now,

And

The Complete Solution

Example:

Solve

Solution:

The auxiliary equation is

The C.F is

And,

Now,

And

The Complete Solution is

Key takeaways

Linear differential equation with constant co-efficient are of the form-

Where are constants.

2. If all the roots are real and distinct, then equation (2) becomes,

3. If two roots are equal

Then complete solution is given by-

4. If one pair of roots be imaginary, i.e. then the complete solution is-

Where and

5. If two points of imaginary roots be equal-

Then the complete solution is-

4.5 The method of successive approximations

Suppose dy/dx = f(x, y) is a first order ODE and with the initial condition y(0) = 0 and if f and are continuous on rectangle R for which the then where y = is the unique solution to this initial value problem.