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

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## Question 1

Appeared in Year: *2016*

### Describe in Detail

Essay▾Find the general integral of the partial differential equation

[CS (Main) Paper 2]

### Explanation

- Given partial differential equation is
Auxiliary equations are

- Integrating we have
- Again each ratio
- Hence general solution of given equation is

## Question 2

Appeared in Year: *2016*

### Describe in Detail

Essay▾Solve the partial differential equation

[CS (Main) Paper 2]

### Explanation

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

## Question 3

Appeared in Year: *2016*

### Describe in Detail

Essay▾Determine the characteristic of the equation and find the integral surface which passes through the parabola [CS (Main) Paper 2]

### Explanation

- 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 …

… (46 more words) …

## Question 4

Appeared in Year: *2015*

### Describe in Detail

Essay▾Solve for the general solution

Where and (CS Paper 2)

### Explanation

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…

… (11 more words) …

## Question 5

Appeared in Year: *2011*

### Describe in Detail

Essay▾Solve the PDF

(CS Paper 2)

### Explanation

Given >

Now P. I

P. I

P. I

And

P. I

So, required solution is

## Question 6

Appeared in Year: *2013*

### Describe in Detail

Essay▾Reduce the equation

to its canonical form when

### Explanation

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)

## Question 7

Appeared in Year: *2015*

### Describe in Detail

Essay▾Solve the Partial Differential equation

Where and (CS Paper 2)

### Explanation

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

… (2 more words) …

## Question 8

Appeared in Year: *2013*

### Describe in Detail

Essay▾Solve

Where D and D՚ denote and . (CS Paper 2)

### Explanation

Given

A. E is

Hence, complementary function

Now,

Hence, the solution of (1) is

## Question 9

Appeared in Year: *2014*

### Describe in Detail

Essay▾Solve the Partial Differential Equation

(CS Paper 2)

### Explanation

Given Differential Equation is

Auxiliary equation is

C. F.

Where being arbitrary functions

Now P. I

Hence, the required solution is

## Question 10

Appeared in Year: *2012*

### Describe in Detail

Essay▾Solve the Partial differential equation

(CS Paper 2)

### Explanation

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.

… (2 more words) …