List of Practice questions on Mathematics III. Unit I: Ordinary Differential Equations. Sub Topic: ...... + is everywher
List of Practice questions on Mathematics III Unit I: Ordinary Differential Equations Sub Topic: Difference Equations Q1. Form difference equation by eliminating the arbitrary constants 𝑎 and 𝑏 from the relations (i) 𝑦𝑛 = 𝑎 cos 𝑛𝜃 + 𝑏 sin 𝑛𝜃 (ii) 𝑦𝑛 = 𝑎𝑛2 + 𝑏𝑛. Q2. Form difference equation from the relation gives as
log(1+𝑧) (1+𝑧)
= 𝑦0 + 𝑦1 𝑧 + 𝑦2 𝑧 2 + … … … +
𝑦𝑛 𝑧 𝑛 . Q3. Solve 𝑦𝑛+1 − 2 cos 𝛼 𝑦𝑛 + 𝑦𝑛−1 = 0. Q4. Solve the difference equation 𝑦𝑚+3 + 16𝑦𝑚−1 = 0. Q5. Solve the difference equation 𝑢𝑛+3 − 2𝑢𝑛+2 − 5𝑢𝑛+1 + 6𝑢𝑛 = 0. Given that 𝑢0 = 1, 𝑢1 = 2, 𝑢2 = 3. Q6. A series of values of 𝑦𝑛 satisfies the relation 𝑦𝑛+2 + 𝑎𝑦𝑛+1 + 𝑏𝑦𝑛 = 0. Given that 𝑦0 = 0, 𝑦1 = 1, 𝑦2 = 𝑦3 = 2; show that 𝑦𝑛 = 2(
𝑛⁄ ) 𝑛𝜋 2 sin . 4
Q7. The integers 0, 1, 1, 2, 3, 5, 8, 13, 21, … … … 𝑛 are said to form a Fibonacci sequence. Form the Fibonacci difference equation and solve it. Q8 . Show that 𝑛 straight lines, no two of which are parallel and no three of which meet in a 1
point, divide a plane into 2 (𝑛2 + 𝑛 + 2) parts. Yet 𝑦𝑛 denote the number of sub regions formed by 𝑛 straight lines. Q9. A plant is such that each of its seeds when one year old produces 8-fold and produces 18fold when two years or more. A seed is planted and as soon as a new seed is produced it is planted. Taking 𝑦𝑛 to be the number of seeds produced at the end of the nth year, show that 𝑦𝑛−1 = 8𝑦𝑛 + 18(𝑦1 + 𝑦2 + … … … + 𝑦𝑛−1 ). Hence show that 𝑦𝑛+2 − 9𝑦𝑛−1 − 10𝑦𝑛 = 0 and solve it. Q10. A sequence of numbers is such that the nth number of the sequence is the sum of twice the (𝑛 − 1)th and three times the (𝑛 − 2)th numbers, where 𝑛 ≥ 2. The first number is zero and second is unity. Find the nth number of sequence. Q11. Solve the difference equation ∆2 𝑢𝑥 + 2∆𝑢𝑥 + 𝑢𝑥 = 2𝑥 . 1
𝑛
Q12. Solve 𝑦𝑛+2 = (2 cos 2) 𝑦𝑛+1 − 𝑦𝑛 + sin 2 .
Q13. Solve 𝑢𝑘+2 + 𝑎2 𝑢𝑘 = cos 𝑎𝑘. Q14. Solve the difference equation 𝑢𝑛+2 − 4𝑢𝑛+1 + 4𝑢𝑛 = 𝑛2 2𝑛 . Q15. A beam of length 𝑙, supported at 𝑛 points carries a uniform load 𝑤 per unit length. The bending moments 𝑀1 , 𝑀2 , 𝑀3 , ………𝑀𝑛 at the supports satisfy the Clapeyron’s equation: 1
𝑀𝑟+2 + 4𝑀𝑟+1 + 𝑀𝑟 = − 2 𝑤𝑙 2 . If the beam weighing 30 kg is supported at the ends and at two other supports dividing beam into three equal parts of 1 meter length (as given in Fig. 1). Show that the bending moments at two middle supports are 1 and 16 units, respectively.
Figure: 1
Q16. Solve the difference Equation: y h3 5 y h 2 8 y h1 4 y h h.2 h
Q17. Solve the difference Equation: u x 2 2mu x 1 (m 2 n 2 )u x m x
Q18. Solve y k 2 6 y k 1 8 y k 3k 2 2 5.3k Q19. Solve E ayk cosnk Apply Generating function technique to solve Q20-Q22: Q20. y(t 1) 3 y(t ) t , Q21. Solve yk 2 2 yk 1 yk 1
y0 1 y0 1 ,
y1 1 .
y0 1 ,
Q22. Solve yk 2 5 yk 1 6 yk 2
y1 1 .
Q23. Express the second order difference equation xt 1 3xt 2xt 1 0 as a system of two first order difference equations and hence solve it with initial values x0 0 , x1 1 by using matrix method.
Solve Q9-Q11 by matrix method z k 1 yk 2z k
Q24. yk 1 2 yk z k ,
Q25. xt 1 4xt 3xt 1 0 with Q26. y k 2 y k sin k 1 2
y1 1 , z1 0
x0 1 , x1 1 y0 1 , y1 1
Q27. Show that the system of three difference equations can be written in the form Vn1 AVn xn Where Vn y n z n
n=0, 1, 2 3 5 2 A 1 1 1 2 1 3
Show further that the system of equation can be written as Vn1 A nV0
n=0, 1, 2
Sub Topic: Legendre and Bessel ordinary differential equations 1. Show that x 0 is not an ordinary point of y x 2 y x y 0. 2. Investigate the nature of the point x 0 for the differential equation x4 y ( x2 sin x) y (1 cos x) y 0 3. Find the first three nonzero terms of each of two linearly independent Frobenius series solution of 2x2 y (sin x) y (cos x) y 0 . 4. Investigate the nature of the point x 0 for the differential equation x3 y x( x sin x) y (1 cos x) y 0 5. Find the power series solution of x2 y 6(sin x) y 6 y 0 . 6. Find the power series solution of differential equation 2x2 y (6x x2 ) y xy 0 . 7. Apply the method Frobenius series solution of Bessel’s equation of order ½, x2 y xy ( x2 1/ 4) y 0, derive the it’s general solution for x 0 ,
cos x sin x . c1 x x 8. Find the general solution of the x2 y xy ( 2 x2 n2 ) y 0 . y( x) c0
9. Evaluate (i) x 2 J 0 ( x)dx (ii)
J ( x)dx
(iii)
3
x J ( x)dx . 3
1
10. Show that 4J n( x) J n2 ( x) 2J n ( x) J n2 ( x) . 11. Show that lim xJ 3/2 ( x) x0
2
.
12. Using Rodrigue’s formula, prove that 13. Prove that Pn (1)
1
P ( x)dx 0 .
1 n
n(n 1) . 2
1
1
1
1
1
1
1
14. Prove that: 𝑃𝑛 (− 2) = 𝑃0 (− 2) 𝑃2𝑛 (2) + 𝑃1 (− 2) 𝑃2𝑛−1 (2) + − − +𝑃2𝑛 (− 2) 𝑃0 (2) 15. Prove that (1 − 2𝑥𝑧 + 𝑧 2 )
−1⁄ 2
is a solution of the equation 𝑧
𝜕2 (𝑧𝑣) 𝜕𝑧 2
𝜕
+ 𝜕𝑥 {(1 −
𝜕𝑣
𝑥 2 ) 𝜕𝑥} = 0 16. If 𝑚 > 𝑛 − 1 and n is a positive integer, prove that 1
∫0 𝑥 𝑛 𝑃𝑛 (𝑥)𝑑𝑥 =
𝑚(𝑚−1)(𝑚−2)−−−−−(𝑚−𝑛−2) (𝑚+𝑛+1)(𝑚+𝑛−1)−−(𝑚−𝑛+3) 1
17. If 𝑚 > 𝑛, show that ∫−1 𝑥 𝑛 𝑃𝑛 (𝑥)𝑑𝑥 = 18. Deduce from Rodrigue’s formula:
2𝑛+1 (𝑛!)2
(2𝑛+1)! 1 ∫−1 𝑓(𝑥)𝑃𝑛 (𝑥)𝑑𝑥
=
(−1)𝑛 2𝑛 (𝑛!)
1
∫−1(𝑥 2 − 1)𝑛 𝑓 (𝑛) (𝑥)𝑑𝑥
19. Prove that: 𝐽0 2 + 2(𝐽1 2 + 𝐽2 2 + − − − − −) = 1 20. Prove that: 𝑥 = 2𝐽0 𝐽1 + 6𝐽1 𝐽2 + − − − + 2(2𝑛 + 1)𝐽𝑛 𝐽𝑛+1 + − −
21. Prove that: 𝐽𝑛 (𝑥) = 𝑑
(𝑥⁄2)
𝑛
√𝜋 𝑔𝑎𝑚𝑚𝑎
𝐽
22. Prove that: 𝑑𝑥 ( 𝐽−𝑛 ) = −
1 (𝑛+ ) 2
1
1
𝑛− ∫−1(1 − 𝑡 2 ) 2 𝑒 𝑖𝑥𝑡 𝑑𝑡
(𝑛 >
2𝑆𝑖𝑛 𝑛𝜋
𝑛
𝜋𝑥𝐽𝑛 2
8
4
23. Show that: 𝐽3 (𝑥) = (𝑥 2 − 1) 𝐽1 (𝑥) − 𝑥 𝐽0 (𝑥) 2
1
3
24. Show that: 𝐽5⁄ (𝑥) = √𝜋𝑥 [𝑥 2 (3 − 𝑥 2 )𝑆𝑖𝑛 𝑥 − 𝑥 𝐶𝑜𝑠 𝑥] 2
25. Show that 𝐶𝑜𝑠 (𝑥 cos 𝜃) = 𝐽0 − 2𝐽2 𝐶𝑜𝑠 2𝜃 + 2𝐽4 cos 4𝜃 − − − − and 𝑠𝑖𝑛 (𝑥 cos 𝜃) = 2[ 𝐽1 𝐶𝑜𝑠 𝜃 − 𝐽3 cos 3𝜃 − − −−]
−1 2
)
UNIT-2 Problem set 1 1. Eliminate the arbitrary constants to obtain a partial differential equation (i) (𝑥 − 𝑎)2 + 𝑦 2 + (𝑧 − 𝑏)2 = 16 (ii) 𝑧 = (𝑥 + 𝑎𝑦)2 + 𝑏𝑦 2. Eliminate the arbitrary function to obtain a partial differential equation 𝑥𝑦𝑧 = 𝑓(𝑥 + 𝑦 + 𝑧) 3. Eliminate the arbitrary function to obtain a partial differential equation 𝑧 = 𝑓(𝑥 + 𝑖𝑦) + 𝑔(𝑥 − 𝑖𝑦), 𝑖 2 = −1. 4. Find the differential equation of all planes which are at a constant distance ‘a’ from the origin. 5. Find the differential equation of all spheres whose centres lie on the z-axis. 6. Find the differential equation of all spheres of radius 3 units having their centres in the xy-plane. 7. Form partial differential equation by eliminating the arbitrary function from the given equation 𝑦 𝑧 = 𝑓( ) 𝑥 8. Form the partial differential equation by eliminating arbitrary constants: 1 𝑧 = 𝑎𝑥𝑒 𝑦 + 𝑎2 𝑒 2𝑦 + 𝑏 2 9. Form the partial differential equation by eliminating arbitrary constants: 𝑧 = 𝑥𝑦 + 𝑦√𝑥 2 − 𝑎2 + 𝑏 10. Form partial differential equation by eliminating the arbitrary function from the given equation 𝑓(𝑥𝑦 + 𝑧 2 , 𝑥 + 𝑦 + 𝑧) = 0 11. Form the partial differential equation by eliminating arbitrary constants: 2
𝑧 = 𝐴𝑒 −𝑝 𝑡 cos(𝑝𝑥) 12. Form the partial differential equation by eliminating arbitrary constants: 𝑒
1 𝑥2 ) 𝑦
𝑧−(
=
𝑎𝑥 2 𝑏 + 𝑦2 𝑦
Problems set-2
1. Form the partial differential equation (by eliminating arbitrary functions) from a.
z ( x y) ( x 2 y 2 )
b. z f ( x at )+g ( x at ) c.
z f ( x2 y2 , z xy) .
2. Find the complete solution of the partial differential equation 3. Solve the partial differential equation
p2 1 q qxy .
2 z 2 z 2 z 5 7 y sin x . x2 xy y 2
4. Find the complete solution of the partial differential equation
p 2 q 2 az .
2 2u 2 u c 5. Find the general solution of heat equation . t 2 x2
6. Find the complete solution of the partial differential equation
z px qy tan pq .
7. A uniform rod of length L whose surface is thermally insulated is initially at temperature
T T0 . Its one end is suddenly cooled to T 0 and subsequently maintained at this temperature; the other end remains thermally insulated. Find temperature distribution T ( x, t ) along the rod.
8. Find the complete solution of the partial differential equation
p2 x qz qxy .
u 2u 9. Find the general solution of heat equation . t x 2 10. A uniform string of line density is stretched to tension c 2 and excutes a small transverse vibration in a plane through the undisturbed line of string. The ends x 0 and
x L of the string are fixed. The string at rest, with the point x b drawn aside through a small distance , is released at time t 0 . Find an expression for the displacement y ( x, t ) .
11. Solve the partial differential equation
2 z 2 z 2 z 6 y cos x . x2 xy y 2
L is held fixed at its ends and is subjected to an initial displacement u( x,0) u0 sin( x L) . The string is released from its position with zero
12. A stretched string of finite length
initial velocity. Find the displacement of the string. 13. Find the complete solution of the partial differential equation
p 1 q qz , where p
z z ,q . x y
Problems set-3
1.
4z 4z 0 Ans:: z f1 ( y x) f2 ( y x) f3 ( y ix) f4 ( y ix) x 4 y 4
2.
4z 4z 4z 4z 2 0 x 4 x 3 y xy 3 y 4
Ans:: z f1 ( y x) f2 ( y x) xf3 ( y x) x 2 f4 ( y x) 3.
1 2z 2z 2z 2 2 e 3x2 y Ans:: z f1 ( y x) xf2 ( y x) e 3x 2 y 2 25 xy y x
4.
( x y) 3 2z 2z 2z 3 2 2 x y Ans:: z f1 ( y x) xf2 ( y 2x) 36 xy x 2 y
5.
2z 2z 2z 2 sin(2x 3y) Ans:: z f1 ( y x) xf2 ( y x) sin(2x 3y) xy y 2 x 2
2z 2z cos mx cos ny 30(2x y) x 2 y 2 1 cos mx cos ny ( 2 x y) 3 Ans:: z f1 ( y ix) f2 ( y ix) 2 2 m n 6.
7.
3z 3z 3z 4 4 4 sin(2x y) x 3 x 2 y xy 2
Ans: z f1 ( y) f2 ( y 2x) xf3 ( y 2x) x 2 cos( 2x y) 8.
2z 2z x 1 sin x cos y Ans: z f1 ( y) f2 ( y 2 x) cos( x y) sin(x y) 2 xy 2 4 x
9.
D
10.
x6 y3 x9 3z 3z 2 3 3 z f ( y x ) f ( y ω x ) f ( y ω x ) Ans: x y 1 2 3 120 10080 x 3 y 3
11.
2z 2z 2z 6 y cos x Ans: z f1 ( y 2x) f2 ( y 3x) y cos x sin x x 2 xy y 2
2
DD' z cos 2y(sin x cos x) 1 1 Ans: z f1 ( y) f2 ( y x) cos( x 2 y) sin(x 2 y) cos( x 2 y) sin(x 2 y) 2 6
2z 2z 2z 2 y 1e x Ans: z f1 ( y x) f2 ( y 2x) ( y 2)e x 2 2 xy x y 13. D D'1D 2D'2 z 0 Ans: 12.
2z 2z 2z 3z x 2 y 1 e cos x 4 4 sin(2 x y) cos mx cos ny 30( 2 x y)x ye 3x 2 y x 2 xy y 2 xy 2
x6 y3 x9 z e f1 ( y x) e f2 ( y x) ( y 2 )e y cos x sin x f3 ( y ω x) cos( 2 x y)f4 ( y x) 120 10080 D D'1D 2D'2 z 0 cos 2 y(sin x cos x) x
2x
x
2
Problems set-4
1. Use the method of separation of variables to solve the equation that u( x,0) 6e x .Ans: u( x , t ) 6e 3 x2 t
2. Use the method of separation of variables to solve the equation
Ans: u(x, y) Ae
1
1 p 2 x
Be1
1 p 2 x
e
u u 2 u ,given x t
2u u u 2 0. 2 x x y
p2 y
2 2y 2 y 3. Transform the equation 2 c to its normal form using the transformation t x 2 u x ct , v x ct and hence solve it. Show that the solution may be put in the form y 1 y f( x ct ) f( x ct ) . Assume the initial conditions y f(x) and 0 at t=0. 2 t Ans: y φ(x ct ) ψ(x ct )
4. Reduce the equation u xx 2u xy u yy 0 to its normal form using the transformation
v x, z x y and solve it. Ans:
2u 0 , u xf1 (x y) f2 (x y) v 2 Wave Equation
2y 2y 1. Show how the wave equation c can be solved by the method of separation x 2 t 2 2
of variables. If the initial displacement and velocity of a string stretched between x=0 and x=l are given by y f( x) and
y g( x) . Determine the constants in the series t
nπct nπct nπx where b n sin sin L L L 1 L L 2 nπx 2 nπx a n f( x) sin dx and b n g(x) sin dx . Here L=l. L0 L nπc 0 L 2. Find the deflection y( x, t ) of the vibrating string of length π and ends fixed, corresponding to zero initial velocity and initial deflection f(x) k(sin x sin 2x) , given Ans: y( x , t )
solution.
a
n
cos
c 2 =1.Ans: y(x, t ) k(cos t sin x cos 2t sin 2x)
3. A string is stretched and fastened to two points l apart. Motion is started by displacing x the string from the initial deflection y A sin , from which it is released at time l t=0.Show that the displacement of any point at a distance x from one end at time t is given by y A sin
x l
cos
ct l
.
4. If a string of length l is initially at rest in equilibrium position and each of its points is y 3 x given the velocity , find the displacement y( x , t ) . b sin l t t 0 bl x ct 3x 3ct Ans: y( x , t ) . 9 sin sin sin sin 12c l l l l 5. A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0, the string is given a shape defined by f(x) x(l x) , μ is a constant and then released. Find the displacement y( x , t ) of any point x of the string at any time t>0. Ans: y( x , t )
8μL2 π3
1
(2n 1) n 1
3
sin
( 2 n 1) πx ( 2 n 1) πct cos L L
Heat Equation
1. A rod of length L with insulated sides is initially at a uniform temperature u 0 .Its ends are suddenly cooled to 0 0 C and are kept at that temperature. Find the temperature
4u 0 function u(x,t) .Ans: u( x, t ) π
(2n 1)πx 1 sin e L n 1 2 n 1
c 2 ( 2 n 1 )2 π 2 t L2
2. An insulated rod of length L has its ends A and B maintained at 0 0 C and 1000 C respectively, until steady state condition prevails. If B is suddenly reduced to 0 0 C and maintained at 0 0 C , find the temperature at a distance x from A at time t. Find also the temperature if the change consist of raising the temperature of A to20 0 C and reducing the temperature of B to 80 0 C .
60x 40 1 2mπx Ans: u( x, t ) 20 sin e L π m 1 m L
4 c 2 m 2 π 2 t L2
3. The initial temperature of an insulated infinite rod is given by u( x,0) ( 1) n U between x=nc and x=(n+1)c where n I Show that for t>0,
2p 1πx 4U 1 u(x, t ) sin e π p 0 2p 1 c
d 2 ( 2 p 1 )2 π 2 t c2
4. A bar with insulated sides is initially at temperature 0 0 C throughout. The end x=0 is u kept at 0 0 C and heat is suddenly applied at the end x=L, so that =A for x=L, where x A is a constant. Find the temperature function u(x,t).
2n 1πx 8AL 1n Ans: u(x, t ) Ax 2 sin e 2 π n 1 2n 1 L 5. Solve the equation
c 2 ( 2 n 1 )2 π 2 t 4 L2
u 2 u with boundary conditions u(x,0) 3 sin nπx , u(0, t ) 0 , t x 2
u(L, t) 0 where 0
