D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
Unit-I
Differential equations of first order and first degree
1) Form the differential equation for the family of all circles of radius 5 with their
centers on X-axis?
2) Find differential equation that represents all parabolas each of which has a
latus rectum 4a and whose axes are parallel to the X-axis?
3) Form the differential equation of all parabolas with X-axis as its axis and  ,0
as its focus.
dy
 3x 2  2 xy  1
dx
dy
5) Solve sin 2 x  y  tan x
dx
dy
6) Solve 2  y sec x  y 3 tan x
dx
4) Solve x
7) Solve xy2  x 2 dx  3x 2 y 2  x 2 y  2 x 3 dy  0
8) Solve x 4 e x  2mxy2 dx  2mx 2 ydy  0
9) Solve 3xy2  y 3 dx  2 x 2 y  xy2 dy  0
10)
Solve
dy
 2 y  e x  x, y (0)  1
dx
11) Find the solution of the differential equation
12) Solve x 2  y 2 
2
dy
 xe y  x and y (0)  0
dx
dy
 xy
dx
dy
 x x2  y2
dx
dy
14) Solve x  y  x 2 y 6
dx
dy
15) Solve  y  y 2 log x
dx
13) Solve x
16) Solve 1  y 2  x  e tan
17) Solve
(i)
1
y
 dy
0
dx
ydx  xdy
 e y dy  0
2
x
(ii)
ydx  xdy
 2 x sin 2 x dx  0
xy
Orthogonal trajectories
1) Find the orthogonal trajectories of the family of circles x 2  y 2  2 fy  1  0.
2) Find the orthogonal trajectories of the family of c sin  ?
3) Find the orthogonal trajectories of the family of curve r n cos  a n .
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
4) Find the orthogonal trajectories of the family of
Circles x 2  y 2  2  y  c  2 , where  is parameter.
5) Find the orthogonal trajectories of the family of parabolas through origin and
foci on Y-axis
6) Find the orthogonal trajectories of the family of curves given by y  Kx 2 , where
K is arbitrary.
7) Find the orthogonal trajectories of the family of the cardiods r  a(1  cos  ) for
different values of ‘a’.
Newton’s law of cooling &Law of natural growth or decay
1) If 30% of a radioactive substance disappears in 10 days, how long will it
take for 90% of it to disappear?
2) If the surroundings are maintained at 30 0 c and the temperature of body
cools from 80 0 c to 60 0 c in 12mins. , find the temperature of body after 24
mints.
3) A pot of boiling water 100 0 c is removed from the fire and allowed to cool a
30 0 c room temperature. Two minutes later, the temperature of the water
in the pot is 90 0 c . What will be the temperature of the water after 5 minutes?
4) A bacterial culture, growing exponentially, increases from 100 to 400 gms in
10Hrs. How much was present after 3 Hrs. from the initial instant?
5) Uranium disintegrates at a rate proportional to the present at any instant. If
M 1 and
1
M 1 grams of uranium are present at times T1 and T2 respectively.
2
Show that the half life of uranium in T1  T2 .
6) A body initially at 80 0 c cools down to 50 0 c in 10 minutes, the temperature
0
of the air being 40 c . What will be the temperature of the body after 20
minutes?
7) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there
are 10000 bacteria. At the end of 5 hours there are 40000. How many
bacteria were present initially?
8) A body is heated to 110 0 c be placed in air at 10 0 c . After 1 hour its
0
0
temperature is 80 c . When Will the temperature be 30 c ?
9) If the air is maintained at 15 0 c and the temperature of the body drops from
70 0 c
0
to 40 c in 10 minutes, what will be its temperature after 30 minutes.
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
UNIT-II
Higher order linear D.E
Rules for finding the complementary function:
Consider the higher order linear equation f(D)y=Q(x) Write
the auxiliary equation f(D)y=0,
Find the roots of auxiliary equation they are
m1, m2 , m3 , m4 ,........., mn .
To find the General solution (C.F) of f (D) y=0:
S.No
1.
Roots of A.E.
f(m)=0
Complementary functions (C.F)
m1 , m2 , m3 ,..........., mn
c1e m1x  c2 e m2 x  ...........  cn e mn x
are rational and
different
2.
3.
4.
5.
6.
Where c1,c2,……….,cn are constants
m1 , m1 , m3 ,..........., mn
( two roots are
rational and equal
and rest are rational
and different)
Two roots of A.E.
are complex say
  i and   i
and the remaining
roots are rational
and different
A pair of conjugate
complex roots
  i are repeated
twice and the
remaining roots are
rational and
different
Two roots of A.E.
are irrational say
   and the
remaining roots are
rational and
different
A pair of irrational
roots    are
repeated twice and
the remaining roots
are rational and
different
(c1  c2 x)e m1x  c3 x m3 x ...........  cn e mn x
e x (c1 cos  x  c2 sin  x)  c3 x m3 x ...........  cn e mn x
e x (c1  c2 x)cos  x  (c3  c4 x)sin  x  c3 x m3x ...........  cnemn x
e x (c1 cosh  x  c2 sinh  x)  c3 x m3x ...........  cnemn x
e x (c1  c2 x)cosh  x  (c3  c4 x)sinh  x   c3 x m3 x  ......  cnemn x
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
find the General solution of f (D) y=Q(x):
ax
Method I: When To Q(x)= e then
yp 
1
eax
f ( D)
 yp 
eax
if f (a)  0.
f ( a)
eax x k
If f (a)  0 write f (D)  (D  a)  (D) where  (a )  0 then y p 
 (a) k !
k
Method II : When Q(x)= sinbx (or) cosbx then
1
sin bx if f (b2 )  0.
sin
bx

y

p
f (D2 )
f (b 2 )
x
if f (b2 )  0 then y p   sin bx dx
2
Similarly for cosbx.
yp 
Method III: When Q(x) = x k , where k is a positive integer then y p 
1
xk
f ( D)
1
1
to the form
by taking out the lowest degree term form f
1   ( D)
f ( D)
1
1
(D). Now we write
as 1   ( D) and expand it in ascending powers of D.
f ( D)
Formulae: i) (1  D)1  1  D  D2  D3  ................
Reduce
ii) (1  D)1  1  D  D2  D3  ................
iii) (1  D)2  1  2D  3D2  4D3  ................
iv) (1  D)2  1  2D  3D2  4D3  ................
Method IV: When Q(x)= eax V ( x) where ‘a’ is a constant and ‘V’ is a function of
x( sin ax or cosax , x k )
yp 
Then
1
eax V ( x)
f ( D)
, y p  eax
1
V ( x)
f ( D  a)
Method V: When Q(x) = xm sin ax (or ) xm co s ax
then y p 
yp 
1
1
x m sin ax  I .P. of
x meiax
f ( D)
f ( D)
1
1
x m cos ax  R.P. of
x meiax
f ( D)
f ( D)
If m=1then y p 

 1
1
1
( xV )   x 
f '( D) 
V
f ( D)
f ( D)

 f ( D)
Method VI: When Q(x) = secx, cosecx, cotx, tanx, etc.
i)
If y p 
If y p 
1
Q( x) then y p  e x  Q( x)e  x dx
D 
1
Q( x) then y p  e  x  Q( x)e x dx
D 
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
Type-I
3
1) Solve D 3  3D 2  4y  1  e  x 
d2y
dy
dy
4
 5 y  2 sin hx subject to y  0 and
 1 at x  0
2
dx
dx
dx
3) Solve D 3  3D 2  4D  2 y  e x
2) Solve

Type-II
1) Solve
2) Solve
3) Solve
4) Solve
5) Solve
6) Solve
7) Solve
8) Solve
9) Solve
10)Solve
Type-III
1) Solve
2) Solve
3) Solve
4) Solve
Type-IV
1) Solve
2) Solve
3) Solve
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D


2
 5D  6 y  sin 4 x sin x
3
 3D  4 y  sin 2 x  7
2
 2D  2 y  cos 9 x
3
 4D y  5  sin 2 x
2
 5D  4 y  2 sin ax
2





 2 y  e  2 x  cos x
2
 3D  2y  sin x sin 2 x
2
 9y  cos(3x  99)
2
 1y  sin x sin 2 x
2
 2 D  2y  e  x  sin 2 x
2

3
 3D 2  10D  24 y  x  3
2
 4D  4 y  8x  e 2 x
3
 6D 2  11D  6 y  e 2 x  x 3
3
 2D  D y  e 2 x  x 2  x  sin 2 x
2
2


2

 1y  x
2

 4 y  x sinh x  54 x  8
2
cosh x
d2y
 9 y  e2x x 2
dx 2
4) Find the general solution of
d2y
dy
 2  y  e x sin 2 x
2
dx
dx
5) Solve D 2  1y  x 2 e 2 x  cos x
6) Solve D 2  2D  3y  x 3 e 2 x
7) Solve D 2  4D  4y  e  x sin 2 x
Type-V
1) Solve D 2  2D  1y  xex sin x
2) Solve D 2  1x  t cos t given x  0,
3) Solve D 2  4D  4y  8x 2 e 2 x sin 2 x
Type-VI
1) Solve D 2  4D  3y  e e
2) Solve D 2  a 2 y  tan ax
x
dx
 0 at t  0
dt
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
UNIT-III
LAPLACE TRANSFORMS
1) Define Laplace transform. State the conditions under which a function is
Laplace transformable. Give some examples.
2) Verify whether the function f (t)  t3 is exponential order and find its
transform.
3) Find the Laplace transform of
(i) e3t  2e2t  sin 2t  cos3t  sinh3t  2cosh 4t  9 (ii) cos3t sin5t
f (t )  cos t,0  t  
4) Find the Laplace transform
f (t )  sin t, t  
1, if 0t 1
0, if 1t  2

5) Find the Laplace transform f (t )  
5, if 2t 3
0, if t 3
6) Find the Laplace transform of
(i) e3t (cos4t  3sin 4t) (ii) e3t (2cos5t  3sin5t) (iii) e3t sin 2 t
7) Find the Laplace transform of 3cos4(t  2) u(t  2)
t e3t sin 2t
t
t
9) Find the Laplace transform of (i) 1 e (ii) 1 e
t
t
10) Using Laplace transform prove that
8) Find the Laplace transform of
 et
(i) 
0
(iii)
sin3t cos t
t

sin 2 t
1
11
dt  log5 (ii)  t 2e4t sin 2t dt 
t
4
500
0
 et  e2t
11) Using Laplace transform evaluate (i) 
t
0
 cos at cos bt

(iii)  tet sin t dt (iv) 
0
0
t
 cos 5t  cos t
dt (ii) 
0
t
dt
dt
1 0t 1
is a periodic function with period 2, find its Laplace

1
1

t

2

12) If f (t )  
transform.
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
13) Find the Laplace transform of periodic function

sin t
f (t )  
 1

0t 



2
t 


with period
2
w
14) A function is periodic in (0,2b) and is defined as
f (t )  1, 0  t  b
. Find the
f (t )  0 , b  t  2b
Laplace transform of f (t).
15) State and prove second shifting theorem of Laplace transform.
16) If L  f (t ) is the Laplace transform, then prove that
 
L  f (t )  s
L f ' (t )  s L  f (t )  f (0) and hence show that
n
n
L f (t )  sn 1 f (0)  sn  2 f ' (0)  ........... f n 1(0)
17) If f(t) is sectionally continuous and of exponential order and if


 
L f (t )  f (s) then L t n f (t )  1
n
dn 
f (s)  .
n 

ds
 f (t )  
   f (s) ds, provided the integral exists.
t

 s
18) If L f (t )  f (s) , then L 
19) Define impulse function (or) Dirac Delta function and find its Laplace
transform.
INVERSE LAPLACE TRANSFORMS
PROBLEMS
1) Find the inverse Laplace transform of

s 3
 s2  6s 13


2
  s 1  s  2  
 e   s  2  

1
3) Find L 

s

2




2) Find L1 
4s  5
 s2 

 3 
4) Find the inverse Laplace transform of cot 1 
2
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology

s 
4
4
 s  4a 
5) Find L1 


3
  s 1 
1
6) Find L1 

1
7) Find
 2 2
2
 s s 1 s  4

2s 3  3

8) Find L1 
2 2
2
 s s 1 s  2
 4  3s 
 e


1
9) Find L 
5 
  s 4 2 


L1 


10) Find






 s 

2
  s  3 
L1 
(i)

s 1

Find L1 
2
 s  2s  2








1
L1 
2
2
 s  5
(ii)



1


1
11) (i)
(ii) L 
2
2 2
2
 s s  a

s 3
12) Find the inverse Laplace transform of
s 2 10 s  29







2




13) State convolution theorem and using it find the inverse Laplace transform
of (i)

s2
 2
2
 s  4 s  9
L1 





1
1 
(
ii
)
L

 2
2

 s 1 s  25






t
dy
 3 y  2 y (t ) dt  t by Laplace transform method.
14) Solve
dt
0
15) Using convolution theorem ,find the inverse Laplace transform of
(i)

1

2 2
  s 1 s  9
L1 




1
 (ii) L1 


2 2

  s  2  s  4



1
 (iii) L1 



2 2
  s 1 s 

D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
d 2x
dx
16) Solve the initial value problem

4
 8 x  e2t . Given that
2
dt
dt
'
x(0)  2 and x (o)  2 .


17) Using Laplace transform, solve D 2  1 x  t cos 2t , given that
x
dx
 0 at t  0 .
dt


18) Using Laplace transform, solve D 2  5D  6 x  5et . Given that
x(0)  2 and x ' (0)  1
19) Solve the initial value problem by using Laplace transform method
y ''  7 y '  10 y  4e 3t , y (0)  0 and y ' (0)  1
20) Solve the differential equation
d2x
dx
 2  5 x  et sin t , x(0)  0 and x ' (0)  1
2
dt
dt
21) By using Laplace transform method solve
D
2
 2 D  2  x  0, given that x  Dx  1 at t  0.
d 2x
22) Solve the differential equation
 9 x  sin t using Laplace transform
dt 2
 
given that x(0)  1, x    1 .
2
23) Solve the D.E y ''  n2 y  a sin  nt  2  , y (0)  0, y ' (0)  0 using Laplace
transform.
24) Solve the following differential equation using the Laplace transform,
y ''  n2 y  a sin  nt  2  , y(0)  0, y ' (0)  0 .
******
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
Unit-IV
Generalized Mean value Theorems
Mean value theorems
x2  y2
2 z
2 z
z
1) If z 
, then prove that x 2  y
2
x y
xy
x
x
2) Using Lagrange’s Mean value theorem, show that

6

1
3 
 sin 1    
2 3
7
4 6
1
3) Using Rolls theorem show that g ( x)  8x 3  6 x 2  2 x  1 has a zero between 0
and 1
4) Expand Tan 1 x in a series of powers of (x-1) up to the term contains the fourth
degree.
5) Obtained the Taylor’s series expansion of sinx in powers of x 
6) Expand log( 1  sin x) in powers of x as the term in x
2

4
6
Taylor’s theorem for two variable functions
1) Find the Taylor’s series for f ( x, y)  e x cos y about the origin up to the terms of
degree 2.
2) Obtain the expansion of e x sin y in powers of x and y.
Jacobian and Functional Dependence
1) If u  x  y  z , uv  y  z , uvw  z show that
2) If u 
zx
(u, v, w)
yz
xy
, v  ,w 
show that
4
x
z
y
( x, y, z )
3) If x  u (1  v) , y  v(1  u ) then prove that
4) If x 
 ( x, y , z )
 u 2v
 (u, v, w)
 ( x, y )
 1 u  v
 (u, v)
u2
v2
 (u , v)
find
, y
v
u
 ( x, y )
5) If u  x 2  2 y , v  x  y  z , w  x  2 y  3z find
(u, v, w)
 ( x, y , z )
6) If u  xy  yz  zx , v  x 2  y 2  z 2 , w  x  y  z , verify whether there exists a
possible relationship in between u, v, w. If so find the relation.
7) Verify whether the functions u  sin 1 x  sin 1 y and v  x 1  y 2  y 1  x 2 are
functionally dependent. If so, find the relation between them.
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
8) If u  x  y  z , u 2 v  y  z , u 3 w  z show that
9) Verify
whether
the
functions u 
 ( x, y , z )
.
(u, v, w)
x y
xz
and v 
xz
yz
are
functionally
dependent. If so, find the relation between them.
Maxima and Minima & Lagrange’s multipliers method
Find the maximum and minimum values of f ( x)  x 3  y 3  3axy.
Find the shortest distance from the point (1, 0) to the parabola y 2  4ax .
Find the minimum value of x 2  y 2  z 2 on the plane x  y  z  3a .
Prove that the rectangular solid of maximum volume that can be inscribed
into a sphere of radius “a” is a cube.
5) Find the maximum and minimum distances of a point on the curve
1)
2)
3)
4)
2 x 2  4 xy  4 y 2  8  0.
6) The temperature T at any point (x,y,z) in the space is given as T  400 x 2 y z.
Find the temperature on the surface of the sphere x 2  y 2  z 2  1.
7) Find the maximum value of f ( x, y, z )  x 2 y 2 z 2 subject to the condition
x2  y2  z 2  c2.
8) Find the maximum and minimum values of x 3  3xy2  15x 2  15 y 2  72 x.
9) Find the minimum value of x 2  y 2  z 2 when ax  by  cz  p.
10) Find the maximum value of f ( x, y, z )  x 2 y 2 z 2 subject to the condition
x2  y2  z 2  c2.
11) Find the minimum value of x 2 y 3 z 4 subject to the condition 2 x  3 y  4 z  a.
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D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
UNIT-V
Partial Differential Equations
1) Solve
p  q  x2
2) Solve ap  bq  cz  0
3) Solve p  q  sin x  sin y
4) Solve xp  yq  z
5) Form the partial Differential Equation by eliminating arbitrary function from
z  f  y  .
 x
6) Solve p 2  q 2  z 2
7) Solve p  q  x  y
8) Solve x 2  y  z  p  y 2 z  x  q  z 2 x  y 
9) Form the partial Differential Equation by eliminating arbitrary function from


f x2  y2 , x2  z 2  0
10) Solve z 2  2 yz  y 2  p  xy  zx  q  xy  zx
11) Solve z 2  p 2  q 2   x 2  y 2
12) Find the differential equation of all spheres of radius 8 and having their centers in
the yz plane.
13) Solve z ( x  y)  p x 2  q y 2
14) Solve
x2 y2

z
p
q
15) Find the differential equation arising from  x  y  z , x 2  y 2  z 2   0
16) Solve x 2  yz  p  y 2  zx q  z 2  xy
D. VIJAYAKUMAR Asst. Prof
NRI Institute of Technology
17) Solve x 2  y 2   p 2  q 2   1
18) Form the partial Differential Equation by eliminating arbitrary function from


f x 2  y 2  z 2 , z 2  2 xy  0
19) Solve x 3  3xy2  p  y 3  3x 2 y q  2x 2  y 2  z
20) Solve x  pz 2   y  qz 2  1
21) Form the partial Differential Equation by eliminating arbitrary function from


f x 2  y 2 , z  xy  0
22) Form the differential equations of all planes which are at a constant distance ‘a’
from the origin.
23) Solve y p  2 y x  log q
24) Solve x 2 p 2  y 2 q 2  z 2
25) Form the partial differential equation of family of cone shaving vertex at origin.
26) Find the general solution of xy 2  z p  yx 2  z q  z x 2  y 2 
27) Form the partial differential equation by eliminating constants a,b and c from
x2 y2 z2


1
a2 b2 c2
28) Find the general solution of  y  zx  p  x  yz q  x 2  y 2 .
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