Unit - 4

Systems of linear differential equations

Q1) What is the general linear system of two first order differential equations in two unknown functions?

A1)

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

For example:

Q2) Define the Matrix form of system of linear differential equations.

A2)

Suppose the system of linear differential equations is gives as below

This system is written as

Where

And

Q3) What do you understand by the basic type of system of two linear differential equations?

A3)

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.

Q4) Solve

A4)

Its auxiliary equation is-

Where-

Therefore the complete solution is-

Q5) Find the P.I. Of (D + 2)

A5)
P.I. =

Now we will evaluate each term separately-

And

Therefore-

Q6) Solve (D – D’ – 2 ) (D – D’ – 3) z =

A6)

The C.F. Will be given by-

Particular integral-

Therefore the complete solution is-

Q7) Find the P.I. Of

A7)

Q8) Find P.I. Of

A8)

P.I =

Replace D by D+1

Put

Q9) Solve

A9)

Auxiliary equation

C.F is

  []

   The Complete Solution is  

Q10) Solve

A10)

The Auxiliary equation is   

  The C.F is

P.I

Now,  

   The Complete Solution is

Q11) Solve

A11)

The auxiliary equation is

The C.F is

 [Put ]

   The Complete Solution is

Q12) Solve

A12)

The auxiliary equation is    

The C.F is

Here ,   ,  .    Let

Now, 

  Put

Multiply by in the numerator and denominator

Put  

   The Complete Solution is

Q13) What is the method of successive approximations?

A13)

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.

Now the functions are successive approximations of the unique solution y = .

Hence we begin with , and the other functions can be find by using the formula given below

Q14) By using the method of successive approximation find the functions of the following differential equation

With the initial condition y(0) = 0.

A14)

Suppose , here f is continuous on all of and is continuous on all of , hence the unique solution exists.

Now define

Here we will calculate the three approximations,

Similarly

Now the third approximations will be

Q15) By using the method of successive approximation find the functions of the following differential equation

With the initial condition y(0) = 0.

A15)

Suppose , here f is continuous on all of and is continuous on all of , hence the unique solution exists.

Now define

Here we will calculate the three approximations,

Now

And