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MATHA COLLEGE OF TECHNOLOGY
SUBJECT: DIFFERENTIAL EQUATIONS
QUESTION BANK
Module I
PART-A (3 Marks Questions)
Find general solution of y’’+8y=0
Find general solution of y’’’-y’’-4y’+4y=0
Find general solution of y’’’-y’’-y’-2y=0
Solve 2y’’+2y’+3y=0
Solve y”-y=0.
Solve y”-6y’-7y=0.
Solve y”+6y’+9y=0.
Find the wronskian of y1=cos wx and y2= sin wx.
Solve y”-2y’+y=0 and check whether y1 and y2 in the general solution are linearly
independent or not.
10. Solve y”’+y’=0.
11. Solve y”’+y”-y’-y=0.
12. Solve y”’-3y”-4y’+6y=0.
13. Solve y”’-2y”-y’+2y=0.
14. Solve yv-3yiv+3y”’-y”=0.
15. Find the general solution of y” – 5y’ +6 y = 0.
16. Find the general solution of y” – 6y’ +9y = 0.
17. Find the general solution of y” – 6y’ +25y = 0.
18. Find the general solution of y” = 4y.
19. Find the general solution of y” +y’ = 0.
20. Find the general solution of yiv+y’” +y “= 0.
21. Find the general solution of y”’+– 6y” +11y’+6y = 0.
22. Find general solution of y’’’-5y’’+7y’-3y=0
23. Find the general solution of y”’ – 8y” +37y’-50y = 0.
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PART-B (3 Marks Questions)
24. . Find an ordinary differential for which the given functions are solutions.
(a) ex, e2x, e3x (b) 1,x, cos 2x, sin 2x (c) ex, e-x, cos x, sin x
25. Find the general solution and check your answer by substitution 4 y”-20y’+25y=0.
26. Solve the initial value problem y”+y’-2y=0, y(0)=4, y’(0)=-5.
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27. Solve the initial value problem y”+y’+0.25y=0,y(0)=3.0, y’(0)=-3.5.
28. Solve the initial value problem y”+0.4y’+9.04y=0, y(0)=0, y’(0)=3.
29. Find an ordinary differential equation for which the given functions are solutions. Show
linear independence by using wronskian.
(a) e 0.5x, e-0.5x
(b)ekx, xekx
30. Solve the initial value problem y”’-y”+100y’-100y=0.
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MODULE II
PART-A (3 Marks Questions)
Find the particular integral of (D2+4)y=cos(3x-2)
Solve (D2+5D+6)y=e-x .
Solve D2+3D-10y=2e2x
Solve (D2-2D+1)y=cos 3x.
Solve (D4-2D3+5D2-8D+4)y=ex.
Solve the differential equation (D4-m4)y = sin mx.
Solve (D3-D2-6D)y=x2+1.
Solve (D2-1)y=5x+2.
Solve (D2-D-2)y= 44-76x.
PART-B (7 Marks Questions)
10. Find the particular integral of (D+1)(D-2)2y=e-x+2cosh2x
11. Find the PI of (D3+1)y=3sin(2x+1)
12. Find the PI of (D2-4D+3)y=sin3x cos2x
13. Solve (D3-5D2+7D-3)y=2e2x cos hx.
14. Solve (D2+4D+4)y=e3x+cos 5x
15. Solve (D2-4D+4)y= sin 2x, given that y= 1/8 and Dy=4 when x=0.
Find the value of y, when x = /4.
16. Solve (D3 + 1)y = cos 2x, find dy/dx.
17. Solve (D3+2D2+D)y= e2x +x2+x.
18. Solve (D2+3D+2)y=4 cos2 x.
19. Solve (D2-2D+5)y= e2x sin x.
20. Solve (D2+2D-3)y=e2x sinx.
21. Solve (D2+2)y= 2 ex sin2x.
22. Solve (D2-2D+1)y= x sinx.
23. Solve 3 x2 d2y/dx2 + x dy/dx + y =0.
24. Solve x2 d2y/dx2 - 4 x dy/dx + 6y =x5.
25. Solve x2 d2y/dx2 - 4 x dy/dx + 6y =x.
26. Solve x2 d2y/dx2 + x dy/dx + y =logx.
27. Solve x2 d2y/dx2 - 3 x dy/dx + y = ( logx(sin log x)+1 ) / x.
28. Solve x2 d2y/dx2 - 2 x dy/dx -4 y =x2+2logx.
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29. Solve x3 d3y/dx3 + 2 x2 d2y/dx2 + 2y = 10(x + 1/x).
30. Solve x3 d3y/dx3 + 3 x2 d2y/dx2 + x dy/dx + y = x+ log x.
31. Solve x2 d2y/dx2 - 2 x dy/dx +2 y =x2+ sin (5 log x).
32. Solve (x2D2-xD+4)y= cos(log x) + x sin (log x).
33. Solve (D2-4D+ 4)y=8x2 e2x sin 2x.
34. Solve (x4D4+6x3D3+9x2D2+3xD+1)y= (1+logx)2.
35. Solve x2 d2y/dx2 - 2 x dy/dx - 4y =x4.
36. A body executes damped force vibrations given by the equation d2x/dt2+ 2k dx/dt + b2x
= e-kt sin wt. Solve the equation in both cases when w2= b2 – k2 and w2 ≠ b2-k2.
37. Solve (3x+2)2 d2y /dx2 + 5 (3x+2) dy/dx – 3y = x2+x+1.
38. Solve (x+1)2 d2y /dx2 + (x+1) dy/dx = (2x+3)(2x+4).
39. Solve (3+x)2 d2y /dx2 + (3+x) dy/dx + y = 2 cos ( log (3+x)).
40. Solve (5+2x)2 d2y /dx2 - 6 (5+2x) dy/dx + 8y = 6x.
41. Solve (2x+3)2 d2y /dx2 -2 (2x+3) dy/dx – 12y = 6x.
42. Using variation of parameters solve d2y/dx2 + 4y= tan 2x.
43. Using variation of parameters solve y”-2y’+y = ex/x.
44. Use variation of parameters to solve y”+y= sec x.
45. Apply variation of parameter to solve (D2-3D+2)y = ex/(1+ex).
46. Use variation of parameter to solve x2 d2y/dx2 + 2x dy/dx -12 y= x3 log x.
47. Solve by variation of parameter d2y/dx2+ y = cosec x.
48. Solve by variation of parameter x2 y” + xy’-y= x3ex.
49. Solve d2y/dx2 – 4 dy/dx +4 y= 3-sin2x+2e2x
50. Solve (D-2) 2 y= 8(e2x+cos2x+x2-1)
MODULE III
PART-A (3 Marks Questions)
1. Find the Fourier Cosine Series as well as the Fourier Sine Series for f(x) = x2, 0< x ≤
c.
2. Find the half range expansion of the function f(x)= x- x2, in 0 < x < 1.
3. Find the Fourier Series expansion of f(x) = Icos xI defined in -π < x < π.
4. Find the Fourier Series expansion of f(x) = x3 defined in -π < x < π.
5. Find the Fourier Series expansion of f(x) = Isin xI defined in -π < x < π.
6. Find the Fourier Series expansion of f(x) = I xI defined in -π < x < π.
7. Find the Fourier Series expansion of f(x) = x-x2 defined in -π < x < π.
8. Find the Fourier Series expansion of f(x) = e-x defined in –L < x < L.
9. Find the Fourier Sine Series expansion of f(x) = x defined in 0 < x < 2.
10. Express f(x)=x as a Half Range Fourier Sine Series in 0 < x < 2.
PART-B (7 Marks Questions)
11. Find the Fourier Series for the 2- periodic function f(x) = 0,
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-π<x<0
= sinx, 0<x<π
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12. Find the Fourier Series for the 2- periodic function f(x) = 1-X, -π<x<0
= 1+X, 0<x<π
13. Find the Fourier Series for the 2- periodic function f(x) = 0, -1< x < 0
= 1, 0 < x < 1
14. Find the Fourier series for the periodic function f(x) with period 2π , defined as
follows
F(x) = 0 , - π < x ≤ 0
= x , 0≤ x ≤ π
What is the sum of the series at x = 0, ±π, 4π, -5π.
15. Find the Fourier Series of the periodic function with period 2, which is given below
F(x) = -x , -1 ≤ x ≤ 0.
= x, 0≤ x≤ 1.
Hence prove that 1 + 1/32 + 1/52+ …… = π2/8.
16. Find the Fourier Series of the periodic function with period 2, which is defined below
F(x) = πx , 0 ≤ x ≤ 1
= π(2-x) , 1 ≤ x ≤ 2
17. Find the Fourier Series for the function with period 4
F(t) = 0 , -2 < t < -1
= k, -1 < t < 1
= 0, 1< t < 2
18. Find the Fourier Series of the function f(x) = x in the range -π<x<π. Hence show that
1- 1/3 +1/5 -1/7 ….. = π/4.
19. Find the Fourier Series for f(x) = x+ x2 in (-1,1).
20. Find the Fourier Series of f(x) defined in (-π,π) as follows f(x) = π/2+x, -π<x<0
= π/2-x, 0≤x≤π
∞
And hence prove that ∑ 1/(2n-1)2 = π2/8.
n=1
21. Find a Fourier Series representation of f(x) = x sin x, periodic with period 2π, defined
in
(a) 0<x<2π (b) -π<x<π
Hence deduce that 1/1.3 – 1/3.5 + 1/5.7 – 1/7.9 +…..= (π-2)/4.
22. Find the Fourier Series for the function f(x) = x, 0<x<1
= 1-x, 1<x<2
2
2
2
2
Deduce that 1/1 +1/3 +1/5 +…. = π /8.
23. Find the Fourier Series expansion of f(x) = ex over the range -1< x <1. What will be
the expansion when x=2.
24. Find the Fourier Series expansion of f(x) = 1+ IxI defined in -3 < x < 3.
25. Given the function f(x)=x, 0 < x < π
(a) Find the Fourier Cosine Series for f(x) and Fourier Sine Series for f(x).
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26. Find the Fourier series for the periodic function f(x) with period 2π , defined as
follows
F(x) = -π , - π < x ≤ 0
= x , 0≤ x ≤ π
27. Find the Fourier Series for the function f(x) = x, 0<x<π/2
= π-x, π/2<x<π
28. Find the Fourier Series for the 2- periodic function f(x) = -k,
-π<x<0
= k, 0<x<π
Also deduce that 1-1/3 +1/5-….= π/4.
29. Find the Fourier Series for the 2- periodic function f(x) = - π,
-π<x<0
= x, 0<x<π
Also deduce that 1+1/32 +1/52-….= π2/8.
n=∞
30. If f(x)= ((π-x)/2)2in the range 0 to 2 π. Show that If f(x)= π2/12+∑ (cosnx)/n2
MODULE IV
PART-A (3 Marks Questions)
1. Find the partial differential equation by eliminating arbitrary constants from
z = (x-a)2+(y-b)2
2. Find the partial differential equation by eliminating arbitrary constants from
z= ax+by+a2+b2
3. Find the partial differential equation by eliminating arbitrary constants from
z = (x+a)(y+b)
4. Find the partial differential equation by eliminating arbitrary constants from
z=(x2+a)(y2+b)
5. Find the differential equation of all spheres of fixed radius having their centre in the
xy – plane.
6. Find the differential equation of all spheres whose centre lies on z axis.
7. Find the differential equation of all planes which are at a constant distance from the
origin.
8. Find the partial differential equation by eliminating arbitrary functions from z=f(x 2-y2)
9. Find the partial differential equation by eliminating arbitrary functions from z = f(x/y)
10. Find the partial differential equation by eliminating arbitrary functions from
z = f(x+ay) + g(x-ay).
11. Find the partial differential equation by eliminating arbitrary functions from
z = f(x+4t) + g(x-4t).
12. Find the partial differential equation by eliminating arbitrary functions from z =
f1(x)f2(y).
13. Form a partial differential equation by eliminating the arbitrary function from
xyz = ф(x+y+z).
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14. Form a partial differential equation by eliminating the arbitrary functions from
z = x g(y) + yf(x).
15. Form a partial differential equation by eliminating the arbitrary function from
x2+y2+z2 = f(x,y).
16. Form a partial differential equation by eliminating the arbitrary function from
f(x+yz, x2+y2-z2)=0.
17. Form a partial differential equation by eliminating the arbitrary function from
z= xyf(x2+y2+z2).
PART-B (7 Marks Questions)
2
2
2
18. Solve y zp+x zq=xy .
19. Solve py+qx = xyz2(x2-y2).
20. Solve (y2z/x) p + x2zq = xy2.
21. Solve p√x+q√y=√z.
22. Solve yzp+zxq=xy.
23. Solve xp+yq=3z.
24. Solve x2(y-z)p+y2(z-x)q = z2(x-y).
25. Solve (y2+z2)p-xyq+xyz=0.
26. Solve xy dx+y2 dy = xy – 2x2.
27. P-2q=3x2 sin (y+2x).
28. Solve x ∂u/∂x +y ∂u/∂y + z∂u/∂z = xyz.
29. Solve (D3 – 4D D’+ 4DD’2)z=0.
30. Solve (4D2+12 DD’+9D’2) z = 0.
31. Solve ∂4z/∂x4 - ∂4z/∂y4 = 0.
32. Solve (D2D’- 4DD’2+4D’3)z=0.
33. Solve ∂2z/∂x2 = a2z given that ∂z/∂x = a sin y and ∂z/∂y = 0 when x = 0.
34. Solve (D3-6D2D’+11DD’2-6D’3)z = e 5x+6y.
35. Solve (D-2D’)(D-D’)2 = e x+y.
36. Solve r – 2s + t = sin (2x+3y).
37. Solve (D2-5DD’+4D’2)z = sin (4x+y).
38. Find the particular integral of (D3-10D2D’+D’3)z = cos(2x+3y).
39. Solve ∂3z/∂x3 - 4∂3z/∂x2∂y + 4 ∂3z/∂x∂y2 = cos (2x+y).
40. Solve r+s-2t=ex+y
41. Solve r+s-2t=√(2x+y)
42. Solve (D2+3DD’+2D’2)z = 2x+3y.
43. Solve (D2-2DD’+D’2)z = tan(x+y).
44. Solve (D2-2DD’-15D’2)z = 12xy.
45. Solve r+(a+b)s+abt=xy
46. Solve (D2+DD’-6D’2)z = ysin x
47. Solve (D2-DD’-2D’2)z = (y-1)ex.
48. Solve (D2D’-2DD’2+D’3)z = 1/x3.
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49. Solve ∂2z/∂x2 - ∂2z/∂x∂y -2∂2z/∂y2 = 2x2+xy-y2 (sinxy-cosxy)
50. Solve (D2+DD’-6D’2)z = ycos x
MODULE V
PART-A (3 Marks Questions)
1. Solve x ∂u/∂x-2y ∂u/∂y=0 using method of separation of variables.
2. Solve ∂2u/∂x2-2 ∂u/∂x+∂u/∂y =0 using method of separation of variables.
3. Use the method of separation of variables to obtain the solution of c2∂2u/∂x2=∂u/∂t that
tends to zero as t →∞ for all x.
PART-B (7 Marks Questions)
4. Solve the boundary value problem ∂z/∂x+∂2z/∂y2=0 satisfying the conditions z(x,0)=0,
z(x,∏)=0,z(0,y)=4sin3y
5. Find the displacement of a finite string of length L .that is fixed at both ends and is
released from rest with an initial displacement f(x).
6. A tightly stretched string of length L is fixed at both ends .Find the displacement u(x,t)
If the string is given an initial displacement f(x) and an initial velocity g(x).
7. Find the solution u(x,y) of the following differential equations using method of separation
of
variables ux+ uy=0.
8. Find the solution u(x,y) of the following differential equations using method of separation
of
variables ux+ uy=(x+y)u.
9. Find the solution u(x,y) of the following differential equations using method of separation
of
variables y2ux-x2uy=0.
10. Find the solution u(x,y) of the following differential equations using method of
separation of
variables uxy- u=0.
11. Find the solution u(x,y) of the following differential equations using method of
separation of
variables xuxy+ 2yu=0.
12. Solve ∂2u/∂x2=2 u+∂u/∂y using method of separation of variables subject to the
conditions u=0 and ∂u/∂x=1+e-3ywhen x=0 for all values of y.
13. Find the deflection of the vibrating string which is fixed at the ends x=0 and x=2 and the
motion is started by displacing the string in to the form sin3 πx/2 and releasing it with zero
initial velocity at t=0.
14. Determine the solution of one dimensional wave equation ∂2Ø/∂x2-1/c2 ∂2 Ø /∂t2=0
0<x<a, t>0 subject to the conditions
(1) Ø(x.0)=f(x)= x/b, 0≤x<b
(a-x)/(a-b), b<x≤a
(2) ∂Ø/∂t(x,0) = 0, 0<x<a
(3) Ø(0,t) = Ø(a,t) = 0, t≥0.
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15. A homogeneous string is stretched and its ends are fixed at x = 0 and x = L. Motion is
started by displacing the string into yhe form f(x) = u0 Sin πx/L from which it is released at
time t= 0. Find the displacement at any point x and t.
MODULE VI
PART-A (3 Marks Questions)
1. Find the temperature distribution in a rod of length L whose end points are fixed at
temperature 0 and the initial temperature distribution is f(x).
2. A long iron rod with insulated lateral surface has its left end maintained at a temperature
0oC and its right end at x=2 maintained at 100oC .Determine the temperature as a function
of x and t if the initial temperature is u(x.0)=100x
0<x<1
100
1<x<2
3.
4.
5.
6.
PART-B (7 Marks Questions)
The ends A and B of a rod of length L are maintained at temperature 0oC and 100oC
respectively until steady state conditions prevails. Suddenly the temperature at the end A is
increased to 20oC and the end B is decreased to 60oC. Find the temperature distribution in
the rod at time t.
Find the temperature distribution in a rod of length 2m whose end points are fixed at
temperature 0 and the initial temperature distribution is f(x)=100(2x-x2).
The temperature at one end of a bar 50 cm long with insulated sides is kept at 0oC and the
other end is kept at 100oC until steady state condition prevails. The two ends are then
suddenly insulated. Find the temperature distribution.
The temperature at one end of a bar of length L cm long with insulated sides is kept at 0oC
and the other end is kept at 100oC until steady state conditions prevails. The two ends are
then suddenly insulated. Find the temperature distribution in the bar.
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