# Q1. Consider the Airy equation Show that, as x there

**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.