CMR Institute of Technology, Bangalore Department(s): Mathematics Semester: 04 Section(s): E,F & G Engineering Mathematics - II Course Instructor(s): Ms. Bharti Sharma Course duration: Jan. 2016-May.2016

15MAT21

Lectures/week: 06

Question Bank Partial Differential Equations 1. Find the PDE of all planes which are at a constant distance b from the origin. 2. Find the PDE of the family of planes, the sum of whose x,y,z intercepts is equal to unity.

3. Form the partial differential equation by eliminating the arbitrary constants a and b from z  ax n  by n . 4. Find the partial differential equation of all planes cutting equal intercepts from the x and y axes.

5. Form the partial differential equation of all spheres whose centre lies on the zaxis. 6. Form a partial differential equation by eliminating the arbitrary constants a and b in each of the following cases: (a) ax 2  by 2  z 2  1 (b) ax  by  ( 1  a 2  b 2 ) z  1 (c) z 2  ( x  a) 2  ( y  b) 2 (d) log( az  1)  x  ay  b 7. Find the PDE by eliminating the arbitrary function

1   log y  x  (b) z  f ( x  ct )  g ( x  ct ) (c) z  f ( y )  g ( x  y ) (d) z  yf ( x)  xg( y ) (a) z  y 2  2 f 

2 2 (e) f ( x  y , z  xy)  0

 

x z

(f) f  z 2  xy,   0 (g) f ( x  y  z, x 2  y 2  z 2 )  0 .

8. Solve

3 z  18xy2  sin( 2 x  y)  0 . 2 x y

9. Solve :

z  2z  2 cos y when x  0 and z  0 when  sin x cos y , given that y xy

y  n . 10. Solve

z  2z  log( 1  y ) when x  1 and z  0 when  e  y cos x , given that x xy

x  0. z  2z  0 when x  0 and z  0 when y  0. 11. Solve  xy , given that y xy z  2z  2 sin y when x  0 and z  0 when y is an  sin x sin y , given that y xy odd multiple of  / 2 .

12. Solve :

13. Solve

z  2z  2.  4 z  0 given that when x  0 , z  e 2 y and 2 x x

14. Solve

z  2z  2 sin y .  4 z  0 given that when x  0 , z  cos y and 2 x x

z  2z  2 sin x. 15. Solve  4 z  0, given that when y  0 , z  0 and 2 y y 16. Solve

z  2z z  x.   2 z  0, given that when y  0 , z  0 and 2 y y y

17. Derive one dimensional wave and heat equation.