Q1. Consider the Airy equation Show that, as x there are solutions Where are constants. Solution: Airy
Question:
Q1.
Consider the Airy equation
Show that, as x
there are solutions
Where 
are constants.
Solution:
Airy function Ai(x) is a special function and Bi(x) is linearly dependent solution to the differential equation
First we prove that
The asymptotes of Airy function is given by
Ai(x) = 
So, exp(
Now x
Now, x
the solution will be
Where 
are constants.
Q2.
Consider the nonlinear ODE
Confirm that 
is an exact solution. Show that if there is another solution such that y(x)
as 
then
y(x)
,
where 
and C is a constant that cannot be determined from local analysis.
Solution:
We now show that 
is an exact solution of the differential equation.
, 
Again, (
= (
= 
This shows that 
is an exact solution of the differential equation.
Now let other solution of the differential equation be
So, 
Let 
{
C 
F = 
Therefore the another solution is
y(x)
, 
where 
and C is a constant that cannot be determined from local analysis.
Q3.
Consider the behavior of solutions to the inhomogeneous Airy equation
That decay as x
find the first two terms in the asymptotic expansion for y(x) as x
explain why the terms in the asymptotic power series are independent of an initial condition.
Solution:
………………………………………………………(i)
Let us consider y = 
From (i)
Now comparing both sides,
And
, 
, 
Again,
, 
, 
Again,
So, y(x) = 
The first two terms are 
.
The terms in asymptotic power series is independent of initial conditions as it is a homogeneous Airy equation and there is one approximation like 
with non-zero terms 
Q4.
Consider the equation
- Show that all three pair-wise dominant balances are inconsistent as
- Show that y(x)
as 
independent of an initial condition.
Solution:
Let us consider the particular solution 
This is clearly not possible. This shows the pairwise dominant balances are inconsistent as 
(b)
………………………………………………..(i)
Integrating factor of (i) = 
= 
= 
Multiplying (i) by integrating factor and hence integrating, we have
y 
[where c is an integrating constant]
= 
= 
= 
Now, 
as 
And it does not depend on the initial conditions.
Expert Answer:
Q1 Consider the Airy equation Show that as x there are solutions Where are constants Solution Airy f... View the full answer
