Download MTE-08 ASSIGNMENT 1

ASSIGNMENT 1
(To be done after studying Blocks 1 and 2.)
Course Code : MTE-08
Assignment Code : MTE-08/TMA-1/2003
Maximum Marks : 100
1.
Which of the following statements are true? Give reasons for your answers.
i)
ii)
iii)
iv)
v)
d3 y x dy

 (cos 2 x ) y  x 3 , is a non-linear differential equation of order three.
3
dx
dx
d
2

3
y
 dy 
The general solution of the differential equation x  2   y    y 4  0 ,
 dx 
 dx 
must contain two arbitrary constants.
1
If (x) =
is a solution of y + y2 = 0 for x  0, then y = c(x) is also a
x
solution of this equation for any value of the constant c.
Every solution of the ODE (D2 + 1)2 y = 0 is bounded on 1,  .
y y + ( y )2 = 0 cannot be solved by reducing it to a first order differential
equation.
(10)
2
4
2. (a) Find the differential equation of the family of circles of fixed radius r with center on
y-axis.
(3)
(b) Show that y = cx2 + x + 3 is a solution of the IVP x2 y  2x y + 2y = 6, with y (0) =
3, y (0) = 1 on the interval  – ,   . Is this solution unique? Give reasons in
support of your answer.
(4)
(c) Determine whether the following functions are homogeneous. If so, state the degree
of homogeneity.
x3y  x 2 y2
i)
f (x, y) =
( x  8 y) 2
 x2 

ii)
f (x, y) = cos 

x

y


3. Solve the following :
a 2 ( xdy  ydx )
(a) x dx + y dy =
x 2  y2
dy
(b)
= sin (x + y) + cos (x + y)
dx
(3)
(5)
(5)
4. (a) For each of the following differential equations state the region in the xy-plane where
the existence of a unique solution through any specified point is guaranteed by the
existence and uniqueness theorem.
xy
y =
i)
ii) y = (1 – x2 – y2)1/2
2x  5y
y = 2xy / (1+y2)
iii)
iv) y = 3(x+y)–2
ln xy
y =
v)
(5)
1 x 2  y 2
3
(b) Suppose a student carrying a flu virus returns to a college campus of 1000 students.
Assuming that the rate at which the virus spreads is proportional to the product of the
number x of infected students and the number of students not infected, determine the
number of infected students after 6 days. It is given that x(4) = 50 and that no student
leave the campus throughout the duration of the disease.
(5)
5. Solve the following :
dy
(a) sin x
+ 3y = cos x.
dx
dz
z
z
(b)
+ lnz = 2 (lnz)2
x
dx
x
(5)
(5)
6. (a) Find the orthogonal trajectories of the family of curves
parameter.
x2
y2

1,  being the
a 2 b2  
(5)
(b) A certain population is known to be growing at a rate given by the logistic equation
dx
 x (b  ax )
dt
Show that the maximum rate of growth will occur when the population is half the
equilibrium size that is, when the population is b/2a.
(5)
1
7. (a) If y1(x) = x– 1/2 sin x is one solution of Bessel’s equation x2 y + x y + (x2  )y = 0,
4
determine a second solution. Consider the interval 0  x   .
(5)
(b) Transform the equation (px – y) (py + x) = h2p, using the transformation
x2 = u, y2 = v. Identify the transformed equation and hence solve it.
(5)
8. (a) Solve the differential equation x3p2 + x2yp + a3 = 0 and also obtain a singular solution
if it exists.
(5)
(b) The differential equation satisfied by a beam uniformly loaded (W kg/metre), with
one end fixed and the second end subjected to a tensile force-P, is given by
d2y
1
EI

Py

Wx 2 ,
2
2
dx
where E is the modulus of elasticity and I is the moment of inertia. Show that the
dy
elastic curve for the beam with conditions y = 0 and
= 0 at x = 0, is given by
dx
W
Wx 2
P
(
1

cos
h
nx
)

, where n2 =   .
y=
(5)
2
2P
Pn
 EI 
9. Solve the following equations :
(a)
(D – 1)2 (D2 + 1)2 y = sin2 x/2 + ex + x
(6)
(b)
(D5 – D) y = 12ex + 8 sin x – 2x
(4)
10. Apply the method of variations of parameters to solve
(a)
x2 y + x y – y = x2ex
(5)
(b)
y + a2y = cosec ax
(5)
4
ASSIGNMENT 2
(To be done after studying Blocks 3 and 4.)
Course Code : MTE-08
Assignment Code : MTE-08/TMA-2/2003
Maximum Marks : 100
1.
Which of the following statements are true? Give reasons for your answers.
x2 (dx)2 + y2 (dy)2 + z2 (dz)2+ 2xy dx dy + 2yz dy dz + 2zx dz dx = 0 is a first
order, first degree differential equation.
i)
2
ii)
iii)
iv)
v)
 2z  2z   2z 
  0 is a linear differential equation.

x 2 y 2  x y 
u 

PDE pu = ey + sin x  p   is a quasi-linear equation.
x 

The equation (y – z)p + (z – x)q = x – y and z – px – qy = 0 are compatible.
Equation uxx + 2uxy + sin2x uyy + uy = 0 is hyperbolic.
(10)
2. Find the integral curves of the following equations :
dx
dy
dz
 2

2
2
x  y  yz x  y  zx z( x  y)
(3)
dx dy
dz
(b)
 
x y z  a x 2  y2  z 2
(a)
(c)
2
(4)
dx
dy
dz


2
x  y 2 xy z( x  y)
(3)
2
3. (a) Find f (y) so that equation f (y) dx – zx dy – xy ln y dz = 0 is integrable. Also obtain
the corresponding integral using Natani’s method.
(5)
(b) Find the differential equation of the family of surfaces [z ( x  y) 2 , x 2  y 2 ]  0 .
(5)
4. (a) Find the surface which intersects the surfaces of the system z(x+y) = c(3z+1)
orthogonally and which passes through the circle x2 + y2 =1, z = 1.
(7)
(b) Show that the complete integral of z = px + qy – 2p – 3q represents all possible planes
through the points (2, 3, 0).
(3)
5. (a) The ends A and B of a rod 20 cm. long have temperatures 300C and 800C until steady
state prevails. After the steady state has prevailed, the temperature of the ends are
400C and 600C respectively. Find the temperature distribution in the rod at time t.
(6)
(b) Solve the PDE; (3D2 – 2 D2 + D – 1)z = 4ex+y cos(x+y).
(4)
6. Find the general solution of the following equations :
(a) (x2 + 3xy2)p + (y2 + 3x2y)q = 2(x2 + y2)z
(b) (x + 2z)p + (4zx – y)q = 2x2 + y
5
(6)
(4)
7. Solve the following :
(a)
x
z
z z
xy
 y  t  az 
x
y t
t
(6)
(b) (D + D – 1) (D + 2 D – 3)z = 4 + 3x + 6y
(4)
8. (a) Find the complete and singular integral of the equation z = px + qy + c 1  p 2  q 2
(6)
(b) Show that the equation
xp = yq and z(xp + yq) = 2xy are compatible and hence find their solution.
(4)
9. Find complete integrals of the following equations. Also obtain their singular solution if
it exists.
(a) z2 (p2z2 + q2) = 1.
(5)
(b) p3 + q3 = 27z.
(5)
10.(a) A string of length  vibrates which is governed by the equation
utt = uxx , 0  x  , t  0
where u(x, t) represents the displacement at x at time t. Initially it has zero velocity
and the initial displacement is u(x, 0) = sin x – sin 3x.
Solve for u(x, t), 0  x   and t  0. At what time is the displacement u(x, t) zero for
all x?
(8)
2
(b) Solve (2 D  DD  D2  6D  3D)z  0
(2)
6