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DEPARTMENT OF MATHEMATICS & STATISTICS, IIT KANPUR
MTH 203, Sem. I, 2005-06
Assignment 1
1. In each of the following classify the equations as Ordinary, Partial, Linear, Non-linear and
specify the order :
y   x sin y  0
(i) y   y sin x  0
(ii)
(iii)
u x u xy  x 2 u  y 2
(iv) ( y ) 2 / 3  ( y ) 2  y  x (v)*
y   xy  
d2
{ cos( xy )}
dx 2
(vi)
( xy )   xy
2. Obtain differential equation of the family of plane curves, represented by (a, b, c are
constants):
(a) xy 2 1  cy
(b)* cy  c 2 x  5
(c)
y  ax 2  be 2 x
(d)* Circles with unit radius and centre on y- axis.
(e)
y  a sin x  b cos x  b
3.(a) Verify that y  ce  x  x 2  2 x  4 is general solution of y   y  x 2  2 (If every
solution of a first order differential eqn. is obtained from one parameter family of G(x, y, c) =
0, then it said to be ‘General solution of the given differential eqn).
(b) Show that the family of curves x 3  y 3  3cxy is represented by the first order equation:
x(2 y 3  x 3 ) y   y( y 3  2 x 3 ) . It is called implicit solution (why?) of the differential eqn.
(c) Verify that y  cx  c 2 is a solution of y  2  xy   y  0 . Also show that y = x2/4 is also
its solution. Note that we can not obtain the second solution from the given family of curves such a solution is called singular solution. Also y  0 and y = x2/4 are two solutions which
satisfy y(0) = 0.
4.* Verify that y  1 /( x  c) is a one parameter family of solutions of the differential
equation: y   y 2 . Also find particular solutions such that (i) y (0) =1, and (ii) y (0) = - 1. In
both the cases, find the largest interval I of definition of the solution.
5. Verify that y  x 2  c as well as y   x 2  k are solutions of first order D. E. ( y ) 2  4 x 2 .
6.* Consider the equation y   y , x>0, where  is a real constant. Show that (i) if  (x) is any
solution and  (x) =  (x) e x , then  (x) is a constant function. (ii) if  < 0, show that every
solution tend to zero.
7. For the following differential equations, draw several isoclines with appropriate lineal
elements. Hence sketch some solution curves of the differential equation:
y  x
y  x  y
(a)*
(b)
 ax  by  m 
 , ad-bc 0 to a separable form. What happens if
8*. Reduce the eqn. : y   f 
 cx  dy  n 
ad=bc?
NOTE: THE MARKED PROBLEMS ARE TO BE FIRST DISCUSSED IN THE TUTORIAL.
Supplementary problems from “Advanced Engineering Mathematics” by E. Kreyszig (8th
edition) – THESE ARE ESSENTIALLY FOR PRACTICE PURPOSE.
(i)
Problem set 1.1: Page – 8:
9, 11and 22.
(ii)
Problem set 1.2: Page -12:
7, 8 and 16.
(iii)
Problem set 1.3: Page- 18:
7, 9, 10, 11, 17, 22 and 25.
(iv)
Problem set 1.4: Page - 23:
1, 2, 6, 9, 11, 12 and 16.
Reading Material – Example 4 (Sec. 1.1); Examples 2, 3, 4 (Sec. 1.4), Example 1 (Sec. 1.7)
DEPARTMENT OF MATHEMATICS & STATISTICS, IIT KANPUR
MTH 203, Sem. I, 2005-06
Assignment 2
1.* Obtain general solution of the following differential equations:
2
 8x  2 y  1 

y   
(a) ( x  2 y  1)  (2 x  y  1) y   0
(b)
 4x  y  1 
2. Show that the following equations are exact and hence find their general solution:
(a)* (cos x cos y  cot x)  (sin x sin y ) y   0 ;
(b) ( x 2  3 y 2  e  x ) y  2 x( ye  x  y  3x)
2
2
dy
1
d
1
d
( Mx  Ny) ln( xy)  ( Mx  Ny) ln( x / y ) = M ( x, y )  N ( x, y )
. Hence,
2
dx
2
dx
dx
dy
infer that (i) if Mx+Ny =0, then M ( x, y )  N ( x, y )
= 0 --- (*) admits 1/(Mx – Ny) as an
dx
I.F., (ii) if M x - N y = 0, then 1/(Mx +Ny) is an I.F. of (*) .(e.g. x 1 / 2 y 3 / 2  x1 / 2 y 1 / 2 y   0. )
3. Verify that
4.* Show that for the differential equation of the type x a y b (my  nxy )  x c y d ( py  qxy )  0 ,
i.e., (mx a y b 1  px c y d 1 )  (nx a 1 y b  qx c 1 y d ) y   0 , where a, b, c, d, m, n, p, q are
constants (mq  np) , admits an integrating factor of the form x h y k . Hence find general
solution of ( x1 / 2 y  xy 2 )  ( x 3 / 2  x 2 y) y   0.
5.* Show that for the equation (3 y 2  x)  2 y( y 2  3x) y   0 , there exists an integrating factor
which is a function of ( x + y2 ). Hence solve the differential equation.
6. Show that 2 sin y 2  ( xy cos y 2 ) y   0 admits an integrating factor which is a function of x
only. Hence solve it.
7.* Obtain solutions of the following differential equations after reducing to linear form:
(i)
8.
y 2 y' x 1 y 3  sin x
(ii)
y  sin y  x cos y  x
(iii)
y   y ( xy3  1) .
Find the orthogonal trajectories of the following family of curves where a is an arbitrary
constant: (i)* e x sin y  a
(ii)
y 2  ax 3 .
9. Find family of oblique trajectories which intersect family of straight lines y  ax at an angle
of 45 .
10. Show that the following family of curves are self orthogonal.
2
(i)* y  4a( x  a)
x2
y2
(ii) 2 
 1 , 0< a < 1,
a 1  a2
Supplementary problems from “Advanced Engineering Mathematics” by E. Kreyszig (8th
edition) – THESE ARE ESSENTIALLY FOR PRACTICE PURPOSE.
(i)
(ii)
(iii)
Problem set 1.4: Page - 32:
Problem set 1.6: Page – 38:
Problem set 1.8: Page – 51:
10,12, 17, 26, 29, 35.
13, 20, 28, 29, 33, 34, 18
7, 9, 10, 15, 17.
DEPARTMENT OF MATHEMATICS & STATISTICS, IIT KANPUR
MTH 203, Sem. I, 2005-06
Assignment 3
1. Solve the following equations by using the method of variation of parameters:
y  y cos x  (sin 2 x) / 2
(i) xy  2 y  x 4
(ii)*
[ METHOD of variation of parameters -- First find general solution yhom to corresponding
homogeneous eqn., and then using y( x)  u( x) yhom ( x) , find general solution of the given eqn.]
2.* Reccati’s equation: y  p( x) y  q( x)  r ( x) y 2 is non-homogeneous equation, which in
general can not be solved by elementary methods. However, if one of its solutions y   ( x)
is known, one can find general solution by using the transformation: y ( x)  u ( x)   ( x).
Show that u(x) satisfies the Bernoulli type equation: u  [ p( x)  2r ( x) ( x)]u  r ( x)u 2 .
Hence solve, y  x2 y  xy 2  1 . (Observe that y  x is an obvious solution of this eqn.).
3. Show that the solutions of the homogeneous linear equation: y  p ( x) y  0 on an interval
I = [a, b] form a vector subspace of the real vector space of differentiable functions on I.
4.* Solve: y  y  x  1 , y  y  cos 2 x , and hence solve y  y  (cos2 x)  x / 2
5. Let f(x, y) be a continuous function on the rectangle R : x  x0  a, y  y0  b . Show that
(a) every solution of the I.V.P. : y  f ( x, y); y( x0 )  y0 is also a solution of
x
y( x)  y0   f (t , y(t ))dt , and conversely.
x0
(b) There exists M >0 such that f ( x, y)  M for all (x, y) in R.
x
(c)* Let h  min(a, M / b) , and yn ( x)  y0   f (t , yn1 (t ))dt , with y0 ( x)  y0 , then show by
x0
the method of induction that yn ( x)  y0  b for x  x0  h .
6.* Let  (x) be a differentiable function on an interval I containing x = 0, such that
3
1
 ( x)  1  2 ( x);  (0)  1 , show that  ( x)  e2 x  .
2
2
7. Using Picard’s method of successive approximations, solve the following IVP and compare
your result with the exact solutions:
2
y ; y (0)  0
y  2 x ; y (0)  1 ,
(i)
(ii)* y  xy  1; y (0)  0
(iii)* y 
3
8.* Obtain the general solution of the following eqns. ( p = y’)
(i)
(ii)
(iii)
p 2  ( x  e x ) p  xex
y  (1  p) x  p 2
x  y  ln p
(iv)
(v)
p2 x  p  p3  1
p7  p3  p 2  1  0 (vi)
y  p5  2 p  1
9. Apply (i) Euler Method, (ii) Improved Euler Method to compute the values of y(x) at x = 0.2,
0.2, 0.6, 0.8, 1.0 for the IVP: y  xy  y 2 , y(0)  1 . Compare the error in each case with the
exact solution.
10.* Solve y  ( y  x)2/ 3  1. Also show that y = x is its solution. What can be said about
uniqueness of the IVP consisting of the above eq. and y(x0 ) = y0.
Supplementary problems: 1. Derive the condition of orthogonal trajectory for the family of
curves F(r, ,c) =0 in polar form.
From : “Advanced Engineering Mathematics” by E. Kreyszig (8th edition)
(i)
Problem set 1.9: Page - 58: 1, 2, 5, 7, 14, 18, 19.
(iii)
Problem set 19.1: Page – 951: 1, 2, 6, 7.