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
