MAT231: DIFFERENTIAL EQUATIONS
Remedial Class Practice Problems
Unit I: First Order ODE's
1. What is order and degree of a differential equation? Find the order and degree of
𝑑3 𝑦
𝑑2 𝑦
4
𝑑𝑦
(i)𝑑𝑥 3 − 3𝑥 5 (𝑑𝑥 2 ) + 𝑑𝑥 = (𝑙𝑜𝑔 𝑦)1/2.
d 3 y  dy 
 
(ii)
dx 3  dx 
2. Solve:
1
2
0
dy
 ( y  1)( y  2)
dx
3. Solve: 1  x 2 dy  1  y 2 dx  0
4. Solve:
5. Solve:
dy y
  x3
dx x
3
dy 1
7. Solve:
 y  3xe x y 2
dx x
dy
8.
 y tan x  y 3 sec x
dx
9. Solve:  3x 2  4 xy  dx   2 x 2  2 y  dy  0
6. Solve:
10. Find the orthogonal trajectories of family of circles x 2  y 2  cy
11. Compute the orthogonal trajectories of the family of the curves given by y2=cx3, where c
is a arbitrary constant.
a
12. Solve: (i) y  p( x  b) 
(ii) y  px  a 2 p 2  b 2
b
13. Solve:
14. Solve:
15. Solve: Solve:
dy y  x  2

dx x  y  4
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Unit II: Solution for Second and Higher Order ODE’s
d3y
dy
 13  12 y  0
3
dx
dx
4
17. Solve:  D  2 D3  D 2  y  0
16. Solve:
18. Solve:  D3  1 y  3  e2 x  5sin 2 x
19. Solve:  D 2  2 D  5  y  sin 3x
20. Solve: ( D 2  1) y  e x  cos 2 x
d2y
dy
 4x  y  4x2
2
dx
dx
2
d y
dy
22. Solve: x 2 2  x  4 y  x 2
dx
dx
2
d y
dy
23. Solve: (1  2 x) 2 2  6(1  2 x)  16 y  8(1  2 x) 2
dx
dx
dx
dy
24. Solve the simultaneous equations
 3x  y ;
 x y
dt
dt
25. Solve the simultaneous equations Dx  3x  2 y ; Dy  5 x  3 y  0
dx
dy
26. Solve the simultaneous equations
 7x  y  0 ;
 2x  5 y  0
dt
dt
d2y
 y  sec x
27. Solve by the method of variation of parameter
dx 2
d2y
 y  cos ecx
28. Solve by the method of variation of parameter
dx 2
d2y
dy
e3 x
29. Solve by the method of variation of parameter 2  6  9 y  2
dx
dx
x
2
2 x
30. Solve by the method of variation of parameter x y2  xy1  y  x e
21. Solve: 4 x 2
Unit III: Partial differential equations
31. Form the PDE by eliminating arbitrary constant z  ( x  a)2   y  b) 2
32. Form the PDE by eliminating arbitrary constant z  xy  y x 2  a 2  b 2
33. Form the PDE by eliminating arbitrary constant x 2 y2  xy1  y  x 2e x
34. Form the PDE f ( x  y  z , x 2  y 2  z 2 )  0
35. Form the PDE  ( x 2  y 2  z 2 , z 2  2 xy )  0
36. Form the PDE g ( xy  z 2 , x  y  z )  0
37. Solve: zxp  yzq  xy
38. Solve: y 2 p  x 2 q  x 2 y 2 z 2
39. Solve : (mz  ny) p  (nx  lz )q  ly  mx
Department of Mathematics
40. Solve: p 2  q 2  1
41. Solve: pq + p + q = 0
42. Solve: p(1 + q) = zq
43. Solve: p (1  q 2 )  q (1  z )
44. Solve: pq = xy
45. Solve: p  x 2  q  y 2
46. Solve: p + q = sinx + siny
47. Solve: p 2  y 3 q  x 2  y 2
48. Find the complete integral of p (1  q 2 )  (b  z )q  0 by Charpit’s method
49. Find the complete integral of px + qy = pq by Charpit’s method
50. Find the complete integral of pxy + pq +qy = yz by Charpit’s method
Department of Mathematics