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BY ASLAM JAINUL
MATHEMATICS 4
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UNIT 1 - Fourier Series.
PART – A
1. Define fourier series.
If f(x) is a periodic function and it satisfies Dirichlet’s condition then
f(x) =
+ ∑ ( cos sin )
where =
∫ ( )
=
∫ ( ) cos
=
∫ ( ) sin
2. Write fourier series of f(x) in ( c , c+2 ).
f(x) =
+ ∑ ( cos
sin
)
where =
∫ ( )
=
∫ ( ) cos
=
∫ ( ) sin
3. State Dirichlet condition.
In the given Interval,
* f(x) should be well defined and single valued.
* f(x) should have finite number of points of discontinuity.
* f(x) should have only a finite number of maxima and minima.
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4. What is Period? Define Periodic function.
A function f(x) which satisfies the relation f(x + T)=f(x) for all x is called
Periodic function where least +ve integer T is called Period of f(x).
Example :
f(x)=sin x is a periodic function with period 2
f(x)=tan x is a periodic function with period
______________________________________________________________________________________
There are three types in fourier series
* Full Range (0 , 2 )
* Checking Model (- )
* Half Range (0, )
Type I : Full Range :
With Respect to ( 0 , 2 )
f(x) =
+ ∑ ( cos sin )
where =
∫ ( )
; =
∫ ( ) cos
=
∫ ( ) sin
With Respect to ( 0 , 2 )
f(x) =
+ ∑ ( cos
sin
)
where =
∫ ( )
; =
∫ ( ) cos
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=
∫ ( ) sin
5. What is even function ?
A function f(x) is said to be even function if f(-x) = f( x ) .
Example : x² , x sinx , cosx .
6. What is odd function ?
A function f(x) is said to be odd function if f(-x) = - f( x ) .
Example : x , x cosx , sinx .
Type II : Checking Model :
With Respect to ( - )
If f(x) is an even function of x in ( - ) then = 0 .
f(x) =
+ ∑ ( cos )
where =
∫ ( )
=
∫ ( ) cos
If f(x) is an odd function of x in ( - ) then = = 0 .
f(x) = ∑ ( sin ) where =
∫ ( ) sin
With Respect to ( - )
If f(x) is an even function of x in ( - ) then = 0 .
f(x) =
+ ∑ ( cos
)
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where =
∫ ( )
and =
∫ ( ) cos
If f(x) is an odd function of x in ( - ) then = = 0 .
f(x) = ∑ ( sin
)
where =
∫ ( ) sin
7. What is Sine Series ?
Series has only sine term is called Sine Series.
8. What is Cosine Series ?
Series has cosine and constant term is called Cosine Series.
Type III : Half Range :
With Respect to ( )
Cosine Series
f(x) =
+ ∑ ( cos )
where =
∫ ( )
=
∫ ( ) cos
Sine Series .
f(x) = ∑ ( sin )
where =
∫ ( ) sin
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With Respect to ( )
Cosine Series .
f(x) =
+ ∑ ( cos
)
where =
∫ ( )
=
∫ ( ) cos
Sine Series .
f(x) = ∑ ( sin
)
where =
∫ ( ) sin
____________________________________________________________________
9. If f(x) = x² sinx in ( - ) , which term of fourier series will have zero
coefficients ?
Let f(-x) = (-x)² sin(-x) = x² ( - sin x) = - x² sin x
f(-x) = f(x) . It is an Odd function ( )
Constant term and Cosine term has zero coefficients.
10. Obtain the value of for f(x) = ( )
in 0 < x <2
=
∫ ( )
=
∫
( )
=
(
( )
)
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= -
[( ) ( ) ]
= -
[( ) ( ) ]
= -
[ ]
= -
[ ] =
11. Find the Fourier Sine series of f(x) = 1 ; 0 < x <
Half Range Fourier Sine Series :
f(x) = ∑ ( sin )
=
∫ ( ) sin
=
∫ sin
=
(
)
=
[cos cos ]
=
[( ) ]
=
[ ( ) ]
f(x) = ∑ (
[ ( ) ] sin )
____________________________________________________________________________________________
Do Yourself :
12.Is the function ( ) {
odd or even.?
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13.Check whether this function is odd or even f(x) = x - x² in (- ).
14.Write down the Fourier series of f(x) = k in (0 , 2 )
15.Express f(x) = x as a half range sine series in 0 < x <2.
16. If f(x)=|x| is expanded in the fourier series in the – find
17. Find the constant term in the fourier series of f(x) = x² in – .
18. Obtain the half range cosine series for ( ) {
19. If the fourier series of the function f(x) = x + x² in the interval
is
+ ∑ ( ) (
cos
sin )
. Find the value of the infinite series
20.If the fourier series of f(x) = |sin x| in (– ) , find the value of
21.If ( ) {
sin then Fourier Series is f( )
[
]
deduce
22.Find for the function f(x) = x + x³ in (– )
____________________________________________________________________
Complex Form of Fourier Series :
23. Write the complex form of fourier series f(x) in (c , c+2l)
The complex form of fourier series f(x) in (c , c + 2l )
( ) ∑ (
)
Where ∫ ( )
dx.
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* In this type, first find value using the formula by substituting the given f(x)
& then Substitute the value in f(x).
* ‘n’ value varies from – to + .
____________________________________________________________________ Do Yourself :
24.Write the complex form of the fourier series for f(x) = in (- , ).
25.Find the complex form of the fourier series for f(x) = sin x in ( ).
26. Find the complex form of the Fourier Series of ( ) {
RMS Value :
27. Define RMS value of f(x) over an interval (a,b).
The RMS value of f(x) ŷ = √∫ ( ( ))
28.Find the RMS value of y=x² in (– )
ŷ √∫ ( ( ))
a= - and f(x) = x²
After Substituting ŷ √∫ ( )
( )
∫ ( )
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ŷ √
=
Parseval’s Identity:
9 State Parseval’s theorem on fourier constants
If the fourier series of f(x) in (-l , l) converges uniformly to f(x) then
∫ ( ( ))
∑ ( )
.
3 Write the Parseval’s identity for f( ) in 0 x .
Parseval’s identity :
∫ ( ( ))
∑ ( )
Odd Function ( OR ) Half Range Sine Series :
∫ ( ( ))
∑
Even Function ( OR ) Half Range Cosine Series :
∫ ( ( ))
∑
31. If the fourier series corresponding to f(x)=x in the interval (0,2 ) is
+
∑ ( cos sin ) without finding the values of Find the
values of
∑ ( )
.
By Parseval’s identity
∑ ( )
=
∫ ( ( ))
=
∫
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=
(
)
=
(
) =
____________________________________________________________________________________________
Do Yourself :
32.Find the value of ∑ ( )
, if the fourier coefficients of f(x)=x² in
- x are
;
( )
;
33.If f(x)= ∑ sin
in 0<x< , what is the value of ∫ ( ( ))
.?
Previous year 2 mark Questions
* Contain SASTRA University previous year Semester questions
* And all Department CIA Questions.
____________________________________________________________________________________________
34.What is the sum of the fourier series at a point x = where the function f(x)
has a finite discontinuity.
Solution : f(x) = ( ) ( )
35. To what value, the fourier series corresponding to f(x)=x² in (0,2 )
converges at x=0.
Solution : f(0) = ( ) ( )
=
=2 ²
36.At a point of finite discontinuity of a function f(x) , what does the fourier
series represent.?
It represents the value of the function.
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37. State Euler’s formula for Fourier Coefficients
38. Define the value of the Fourier series of f(x) at a point of discontinuity.
39. If f(x) is defined in -3 3 Find the value of .
40. Write down the Fourier series for an odd function f(x) in (- , ).
41. What is harmonic analysis in Fourier Series.
42. State Parseval’s identity in Fourier Series
43. Define odd and even functions graphically.
Find ‘ ’ for f( ) in -2 < x < 2 , f(x + 4) = f(x).
5 Find ‘ ’ for f( ) - x² in the range (0,3).
46. If x=c is a point of discontinuity then the Fourier Series of f(x) at x=c is
given by f(x) = __________.
47. Find the period of the periodic functions y = sin 4x.
48. Find the RMS value of the function f(x) = 2 in (0,3).
49. Find the constant for the function f(x) = 3 in .
50. If f(x) = x² + x³ is expressed as a Fourier series in the interval (-2,2) to
which value this series converges at x=2 ?
51. If f(x) is defined in (0,2 ) , then the value of Fourier series of f(x) at x=0 is
equal to ______.
52. Find the root mean square value of f(x) = x - x² in the interval (-1 , 1).
53. Find the half range sine series of f(x)=x in (0,l).
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54. If the fourier series expansion x² in ( ) is x² =
+
∑ (( )
cos )
, deduce the sum of the series
55. Obtain the half range cosine series for ( ) {
56. Under what conditions, the Fourier series for f(x) in (c,c+2 ) is possible ?
57. Find for the Fourier series expansion of sin(3x) in (- ).
58. If , are the Fourier coefficients in the real expansion of f(x) over (0,2l)
and the coefficients in the complex expansion, what is the connection
between them?
59. What is the advantage of having multiple of 4 equally spaced numerical
values of f(x) in harmonic analysis.
6 The Fourier coefficients of the sum f₁( ) f₂( ) are _____.
61. If the fourier series expansion of ( ) {
is
f(x)=
(cos
) then find the value of ∑
( n
) .
62. Check whether the function f(x)= is odd or even.
63. Is f(x)=tan x has Fourier series in any interval ?
64. If f(x)=sinh x is defined in ( ) , Find the value of .
65. A function f(x) is defined in ( ) and if f(x) = x + 1 in (0, ) , find f(x) in
(- ) if f(x) is odd.
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PART – B
Model 1 : Fourier Series Expansion
1. Obtain the Fourier series expansion of f(x)=|sin x| in .
2. Find the half range Fourier cosine series for f(x)= kx ( ) in ( ).
3. Express cosh(2x) as a Fourier series in .
4. If ( ) {sin
cos
, Express f(x) in a Fourier sine series.
5. Find the Fourier cosine series for f(x)= x ( ) in ( ).
6. Find the Fourier sine series for f(x)= x² in ( ).
7. Find the Fourier series of f(x) = x - x² in .
8. Find the half range cosine series for the function ( ) {
9. Express f(x)= ( )² as a Fourier series in the interval (0,2 ).
10. Obtain cosine series for ( ) {cos
.
11. Expand as a Fourier series in the range 0 to 2 .
12. Expand ( ) {
as a Fourier series of period ‘ ’
13. Find half range sine series for the function ( ) {
14. Find the fourier series of f(x)= ( )² in the interval (- ).
15. Find the fourier series of ( ) in the interval (0 ).
16. Express ( ) as a Fourier series in .
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17. Obtain the sine series for the function ( ) {
.
18. Find the Fourier series to represent f(x)= x² - 2 in the interval (- ).
19. Expand x - x² in a Sine series in ( ) the first three terms.
20. Find the Fourier series of f(x) = x + x² in .
21. Obtain the Fourier Series for f(x) = (x - 1)² in 0 < x < 1.
22. Find the fourier series for ( ) {
.
23. Find the half range Fourier sine series for f(x) = lx-x² in (0,l)
State and prove Parseval’s theorem
25. Find the Fourier series expansion of f(x) {
.
Sometimes problem will be asked like “Find ∑
with the help of f( )”
After finding , , substitute in f(x) and , derive it.
Model 2 : Proof Sums
To find Solution
+ (OR) ∑
+
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+
+
+
3
∑
( )( )
+
+
∑
∑
( )
(OR) ∑
( )
Model 3 : Complex Form.
26. Find the complex form of the Fourier series of f(x)= in (-l,l).
27. Find the complex form of the Fourier series of f(x)= in (0,2 ).
28. Find the complex form of the Fourier series of f(x)= in (-l , l).
29. Find the complex form of the Fourier series of f(x)= in (- , ).
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Model 4 : Harmonic Analysis.
30. The values of x and the corresponding values of f(x) over a period T given
below. Show that f(x) = 0.75 + 0.37 cos + 1.004 sin where
x 0 T / 6 T / 3 T / 2 2T / 3 5T / 6 T
f(x) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
31. Compute the first three harmonic functions of the Fourier series of f(x)
given by the following table :
x 0
3
3
3
5
3
2
f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
32. Obtain Fourier series upto the second harmonic to represent the relation
between x and y to the following data :
X 0
3
3
3
5
3
Y 0.8 0.6 0.4 0.7 0.9 1.1
33. Determine the first two harmonics of the Fourier series for the following
values :
X 0
3
3
3
5
3
Y 1.98 1.30 1.05 1.30 -0.88 -0.25
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34. Obtain the first two harmonic for the data :
X 0 1 2 3 4 5
Y 4 8 15 7 6 2
35. Determine the first 3 harmonics of the Fourier series for the values :
X 30 60 90 120 150 180 210 240
Y 2.34 3.01 3.68 4.15 3.69 2.20 0.83 0.51
X 270 300 330 360
Y 0.88 1.09 1.19 1.64
36. Find the Fourier series upto the third harmonic for the function y = f(x)
defined in (0, ) from the table :
x 0
6
6
3
6
6
5
6
f(x) 2.34 2.2 1.6 0.83 0.51 0.88 1.19
37. Obtain the first two harmonic for the data :
X 0 1 2 3 4 5
Y 9 18 24 28 26 20
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UNIT 2 - Partial Differential Equations.
PART – A
1. Define PDE with example.
An equation which involves partial derivatives of a function of 2 or more
variables called Partial Differential Equation.
Example :
+
= 0.
Define Lagrange’s linear equation
A first order linear PDE which is of the form pP+qQ = R , where P , Q , R are
functions of y z are called Lagrange’s equation
Auxiliary Equation:
=
=
Given a derivative , Find z.
(Note : Integrate the given equation.)
3. Solve :
= xy.
(
) = xy
Integrate w.r to x
(
) =
+ f(y)
Integrate w.r to x
Z =
+ f(y) . x + g(y)
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Do yourself :
4. Solve :
= sin y.
5. Solve :
= sinx.
Elimination of Arbitary constants.
6. z = (x² + a) (y² + b)
( )
( )
Substituting 1 and 2 in given equation.
Z =
4xyz = pq.
Do Yourself :
7. Solve 2z =
+
.
8. Solve (x – a )² + (y - b)² + z² = 1.
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Elimination of function.
9. Form PDE : ( ) = y² + z².
( ) . 2x = 0 + 2z
( ) =
( ) . 2y = 2y + 2z
( ) = 1 +
Comparing both 1 and 2
The solution is
-
= 1
Do Yourself :
10. Solve : y = f (
)
11. Solve : z = f(2x + y)
12. Solve : z= f(x + by). .
Type No 1 - F(p,q) = 0.
13. Solve p³ - q³ = 0.
* It is of the form F(p,q) = 0.
* Assume z = ax + by + c be a solution.
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* After DF w.r to x & y
= a ;
= b ;
* We know that
and
* Hence p = a , q= b;
* a³ = b³ * a =b
* Substitute a or b value in z = ax + by + c.
*Soln is z = ax + ay + c or z = bx + by + c
Do Yourself :
14. Solve p + q = pq.
15. Find complete solution of pq + p + q = 0.
(Note : pq p q is not Caliraut’s form.)
Type No 2 - Caliraut’s form
It is of the form z = px + qy + f(p,q).
16.Find the complete (general) solution of z=px+qy+pq.
* It is of the form z = px + qy + f(p,q).
* For complete solution, replace p=a , q=b.
* The complete Solution is z=ax+by+ab.
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17. Find the singular solution of z=px+qy+√pq.
* General Solution is z=ax+by+ .
* For Singular solution,
then
x +
. b
0 = x +
x =
then
y +
. a
0 = y +
y =
Multiplying x and y , xy =
4xy = 1
Do Yourself :
18. Find the complete solution of (pq-p-q)
Type No 3 - f(z,p,q) = 0.
Steps to be followed:
* z = f(u) ; u = x + ay ;
* p =
and q = a .
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19. Solve : p(1 + q) = qz.
*It is of the form f(z,p,q) = 0.
* Substitute p =
and q = a .
*
(1 + a .
) = a .
z
Canceling common terms and simplifying we wil get.
== a .
= az – 1
∫
∫
log (az-1) =
+ c
Substitute u=x+ay and Simplifying we get (u = x + ay)
log (az-1) = (x + ay) + c
Type No 4 - F₁( p) F₂(y q)
Steps to be followed:
* Equate F(x,p) and F(y,q) with k.
* dz = pdx + qdy
20. Solve py = q.
P =
P =
=k
Then p = k and q = yk.
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dz = pdx + qdy
z ∫pd ∫qdy
z=kx + k
+ c
Do Yourself :
21. Solve pq=xy.
22. Solve p + q = sin x + sin y.
23. Solve p - x² = q + y².
Lagranges Linear Equation - Grouping Method.
* Of the form pP + qQ = R.
*
=
=
24.Solve pyz + qzx = xy.
* It is of the form pP + qQ = R.
Where P=yz , Q=zx and R = xy.
*
=
=
Grouping 1 :
=
x dx = y dy
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-
= u(x,y)
Grouping 2 :
=
y dy = z dz
-
= v(x,y)
Solution is ø(u,v) = 0.
Therefore ø (
-
,
-
) = 0
Do Yourself :
25. Solve: xp – yq = xz.
Homogen. Linear Eqn. with Const. Coeff.
* To find C.F and P.I
Complementary Functions ( C.F ):
* If roots are distinct then C F is f₁(y m₁ ) f₂(y m₂ ) …
* If two roots are equal then C F is f₁(y m₁ ) f₂(y m₁ ) f₃(y m₃ ) …
* If three roots are equal then C F is f₁(y m₁ ) f₂(y m₁ ) f₃(y m₁ )
f ₄(y m₄ ) …
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26. Find C.F of (D² + 1) z = 0.
m² + 1 = 0
m=± i (Distinct)
z f₁(y i ) f₂(y-ix)
Note:
* Don’t worry about comple roots check whether that is equal or not.
* D m D
* Another form of Question: 4
- 12
+ 9
= 0.
Where
and
Do Yourself :
27. Find C.F of (4D² - DD 9D ) z
28. Find C.F of (D³ - 6D D DD - D ) z = 0.
29 Find C F of (D DD 3D ) z
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Particular Integral:
* When F(x,y)=eᵃˣ⁺ᵇʸ and P.I =
( ) eᵃˣ⁺ᵇʸ
Substitute D a D b and simplify
In case if f(a,b)=0 , Differentiate denominator and multiply Numerator
with x.
* When F(x,y)= sin(ax + by) or cos (ax + by) and
P.I =
( ) sin(ax + by) or cos (ax + by)
Substitute D² = -a DD - ab D -b² and simplify.
* When F(x,y)= xᵐyᵖ and
P.I =
( ) xᵐyᵖ [ f(D D’) ] ⁻ xᵐyᵖ
E pand [ f(D D’) ] ⁻ in ascending powers of
and neglect higher terms.
Note :
* In Book they used x power m and y power n , instead of n we used p.
Required Formula :
*
……
*
= 1 - x + x² - ……
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30. Solve: (8D² – 3DD D ) z eˣ⁺ʸ
D a ; D b
P . I =
eˣ⁺ʸ
=
eˣ⁺ʸ
31. Solve : (D 3 DD D ) z = sin (x + 5y)
D² = -a² = - and D -b² = -25
P.I =
(sin x + 5y)
=
(sin x + 5y)
=
(sin x + 5y)
Do Yourself:
32. Solve: (D² – DD 3D ) z eˣ⁺ʸ
33. Solve: (D DD 3D ) z = cos (2x + y).
34. Solve: (D D ) z eˣ⁻ʸ
35. Solve: (D² - D ) z = xy.
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Previous year Questions
* Contain SASTRA University previous year questions
* And all Department CIA Questions.
____________________________________________________________________________________________
36. Solve:
6 3 ;
3
37. Solve: x²p + y²q = z².
38. Find the particular integral of (D²- DD 3D ) z .
39. Form the PDE from ( ) by eliminating .
40. Find the general solution of
-12
+ 9
= 0
41. Find the complete integral of the PDE , p-x² = q+y².
42. Solve
+ z = 0.
43. Eliminate ‘a’ and ‘b’ from z
+
and form the PDE.
44. Solve:
+ xzq = y².
45. Find the PDE of the family of spheres having their centers on the line x=y=z.
46. Solve
= sin x.
47. Find the complete integral of the PDE z=px+zy+pq.
48. Form PDE by eliminating the arbitrary constants in z=(x-a)²+(y-b²)+1.
9 Solve (D - D’ )z
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5 Find the particular integral of (D DD’) z .
51. Find the differential equations of all planes through the origin.
52. Obtain the partial differential equation of all spheres of radius ‘C’ having
their centers on the xy-plane.
53. Solve : (1-x)p + (2-y)q = 3-z.
54. Define complete integral of PDE.
55. Form PDE by eliminating arbitrary function f from z = f ⟨
⟩
56. Find the complete integral of pq=5.
57 Solve ( DD’ – D’ )z
58. Form PDE by eliminating a and b from z=a +b .
59. By eliminating ‘f’ from z f(2x+y) , form PDE.
60. Find the complete solution p+q=sinx + siny.
61. Solve : - xp + yq = z.
6 Solve (D D D’-DD’ -D’ ) z
63. Solve p(1+q)=qz.
6 Form PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from
z=(x+a)(y+b).
65. Solve py=q.
66. Find the particular Integral of
- 5
+6
=
67 Define Lagrange’s linear equation and write its auxiliary equation.
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68. Find the particular integral of (D² - D’ ) Z y
69. Write the form of Clairaut’s equation in partial differential equations
70. Write the geometrical interpretation of singular integral.
71. Write down the suitable substitutions to reduce p²+q² = z²x² + z²y² into the
standard form of partial differential equation.
72. Form the partial differential equation by eliminating the arbitrary function
(
).
73. Define complete integral.
74. Solve pq=1.
75. Write down the suitable substitutions to reduce qx²+py² = zpq into the
standard form of partial differential equation.
76. Find complementary function of (D² + 4D +3)z = sin h(3x-2y).
77 Form PDE by eliminating arbitrary constants ‘l’ and ‘m’ from z=lx³+my³.
PART – B
Model 1 : Homogeneous Linear Equation with Constant Coefficients.
To find CF and PI
____________________________________________________________________________________________
Question may be in the form of :
- -6 = y cos x
(D² - DD’ -6D’ ) ycos
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- s - 6t = y cos x
-
-6
= y cos x
1. Solve: + -6 = y cos x. (OR) (D² - DD’ -6D’ ) ycos .
2. Solve: - -2 = ( y - 1) .
3. Solve: (D³ - 7DD’ - 6D’ ) z y
Solve: (D D D’ - DD’ - D’ ) z cos ( y)
5. Solve: (D DD’ D’ ) z y .
6. Solve: (9D 6DD’ D’ )z ( + )².
7. Solve:
- 5
+ 6
= sin 4x cos 3y.
8. Solve: (D³ - 7DD’ - 6D’ ) z sin ( y) .
9. Solve: (D² - 5DD’ 6D’ ) z y sin
10. Solve: (D² - DD’) z y .
11. Solve: (D² - D’ ) z sin(2x + 3y).
Solve: (D 3DD’ - D’ ) z sin y
13. Solve: (D² - DD’) z cos cos y
Solve: (D D D’ -DD’ -D’ ) z cos 2y.
15. Solve:
- 4
+ 4
= .
16. Solve:
- 2
= + 3x²y. (OR) (D³ - D D’) z 3 y sin ( y)
17. Solve : (D² + 4DD’ - 5D’ )z 3 + y² + x + .
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18. Solve : (D² - 3DD’ D’ ) z sin cos y
9 Solve : (D DD’ - 3D’ )z cos ( 3y)
Solve : (D 3DD’ D’ ) z y
Model 2 : Lagrange’s Linear Equation : Multiplier
21. Solve: (z² - 2yz -y²)p + (xy + zx)q = (xy - zx).
22. Solve: x(y²+z)p – y(x²+z)q = z(x²-y²).
23. Solve: x(z²-y²)p + y(x²-z²)q = z(y² - x²).
24. Solve: (mz-ny)p + (nx - lz)q = ly-mx.
25. Solve: (x²+y²+yz)p + (x² + y² -xz)q = z(x+y).
26. Solve: x(y-z)p + y(z-x)q = z(x-y).
27. Solve: (y²-z²)p + (x²-z²)q = (x²-y²).
28. Solve: (3z-4y)p + (4x-2y)q = 2y-3x.
29. Solve: (x² - yz)p + (y²-zx)q = z²-xy.
30. Solve: (x²-y²-z²)p + 2xyq = 2zx.
Model 3: Clairaut’s Form
31. Solve: z=px + qy + p² + q² + pq.
32. Solve: z=px + qy + c √p q .
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33. Solve: z=px + qy -2 √
34. Solve: z=px + qy + p²q².
35. Solve: z=px + qy + p² + q².
36. Solve: z=px + qy + √p q .
37. Solve: z=px + qy + √p q 6 .
Model 4 : F(z,p,q)=0.
38. Solve: 9(p²z+q²) = 4.
39. Solve: z² (p² + q² + 1) = a².
40. Solve: z² = 1 + p² + q².
Solve: 9pq z ( z )
Model 5 : Elimination of arbitrary constants & functions.
Form PDE by eliminating ‘f’ and ‘g’ from z f (
)+ y g(x).
43. Obtain PDE by eliminating f from xy + yz + zx = f (
).
Form PDE by eliminating arbitrary functions ‘f’ and ‘g’ from z f( y)
g(3x-y).
Model 6: It is of the form : (x,p) = (y,q)
45. Solve: p² + q² = x + y.
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46. Solve: p²y (1+x²)=qx².
Model 7 : F(z, ) = 0 & F( ) = 0
47. Solve: x²p² + y²q² = z².
48. Solve: x²p + y²q = z².
49. Solve: p² + x²y²q² = x²z².
50. Solve: p²z²sin² x + q²z²cos² y = 1.
5 Solve: p y zq z
Model 8 : Other Problems.
52. Solve: x
3
53. Solve
= sin x sin y given that
sin when x=0 and z=0 when y is
an odd multiple of
.
54. Find the equation satisfying the equation xp - yq=z and passing through the
circle x² + y²=1 , z=1.