Partial Differential Equations [Optionals IAS Mains Mathematics]: Questions 1 - 10 of 29

• Given partial differential equation is

  Auxiliary equations are

• Integrating we have

• Again each ratio

• Hence general solution of given equation is

Given partial differential equation is

S. F. is

A. F. is

Complementary function is

Particular integral

Where

Case fail denominatar

Particular integral

where

General solution is

• The given differential equation is

• The initial condition for are

• The initial condition of are determined by

  and

  and

  and

  and

  and

• The characteristic equations are

• Integrating, we have

• Appling Initial condition

  we get

• Also

• On integrating, we have

• Appling initial condition

  We have

  Also

• On integrating, we get

• Apply initial condition

  Now

• on integrating both …

We have

The Lagrange՚s auxiliary equations are

Choosing as multipliers, each fraction of (1)

Choosing as multipliers, each fraction of (1)

From (1) , (2) , and (3)

Taking the first two fraction of (4)

Putting

(5) reduces to

Integrating

Taking the two fraction of (4)

On R. H. S of (7) , Putting

Then (7) Reduces to

On Integrating, we get

From (6) and (8) , t…

Given >

Now P. I

P. I

P. I

And

P. I

So, required solution is

Given

Comparing (1) with

Here,

Equation reduce to

Then, the corresponding equation are given by,

Integrating, these

In order to reduce (1) to its canonical form, we choose

Now,

Also,

Using (5) , (6) and (7) in (1) , we get

It is the required canonical form of (1)

We have, Partial differential Equation

Where and

The Lagrange՚s auxiliary equation for the given equation are

Taking last two fractions of (1) , we get

Now

Combining the third fraction of (1) with fraction (3) , we have

From (2) and (4) solution is

Where being an arbitrary function

Given

A. E is

Hence, complementary function

Now,

Hence, the solution of (1) is

Given Differential Equation is

Auxiliary equation is

C. F.

Where being arbitrary functions

Now P. I

Hence, the required solution is

Given,

Lagrange՚s auxiliary equation for the given equation are,

Taking the first and last fraction of (2) ,

Taking second and last fraction of (2) , we have

From (3) and (4) , the required general solution is

being an arbitrary function.