Questions and Answers

PART-A

QUESTIONS AND ANSWERS

1. Explain how p.d.e are formed.

2. Form the p.d.e by eliminating the arbitrary constants a and b from z = ax + by.

[Ans. z = px + qy]

3. Eliminate the arbitrary constants a and b from

z = ax + by + a2 + b2.

[Ans. z = px + qy + p2 + q2]

4. Find the p.d.e. of all planes through the origin.

[Ans. z = px + qy]

5. Form the p.d.e by eliminating the arbitrary function from

z = f(x2 – y2).

[Ans. py + qx = 0]

6. Obtain p.d.e. from z = f (sin x + cos y)

[Ans. p siny + q cos x = 0]

7. Solve :

 [Ans. z = cos x + f (y)]

8. Define general and complete integrals of a p.d.e.

9. Define complete integral and singular integral.

10. Solve: √p + √q = 1.

[Ans. z = ax + (1 − √a)2y + c]

11. Solve p = 2qx 

[Ans. z = ax2 + ay + b]

12. Solve z = px + qy + p2

[Ans. z = ax + ay + a2]

13. Find the complete integral of (p − q) (z - px - qy) = 1

[Ans. z = ax + by + 1/a-b] 

14. Find the complete integral of pq = xy

[Ans. 2az = a2x2 + y2 + b]

15. Solve px + qy = z

[Ans. ]

16. Solve : (D2 – 4DD' + 3D '2) z = 0

[Ans. z = ϕ1 (y + x) + ϕ2 (y + 3x)]

17. Find the P.I. of { D2 + 4DD ' ] z = ex. 

[Ans. P.I  = ex]

18. Find the P.I of (D2 + DD ') z = ex-y

[Ans. P.I = xex-y]

19. Solve : [D2 - 2DD' + D'2]z = cos (x - 3y)

[Ans. z = f1 (v + x) + xf2 (y + x) − 1/16 cos (x - 3y)]

20. Solve : [D2 - DD' + 6D'2] z = 0

[Ans. z = ϕ1 (y − 2x) + ϕ2 (y + 3x) ]

21. Solve : 2r + 5s - 3t = 0

[Ans. z = ϕ1 (y - 3x) + ϕ2 (2y + x)]

22. Find the general solution of p tanx + q tan y = tanz

[Ans.  ]