Engineering Mathematics 1 (010.140)

Engineering Mathematics 1 (010.140) nd. 2 Midterm Examination. #1. Advanced engineering mathematics, KREYSZIG, 9th, 4.5-Ex.1 A pendulum consisting of a body of mass m (the bob) and a rod of length L. Assume that the mass of the rod and air resistance are negligible. (1) Set up the mathematical model. Sol). θ +ksinθ=0. (2) Determine the locations and types of the critical point. Sol). ① Critical points (0, 0), ± (2π, 0), ± (4π, 0), … , Linearization. →(0,0) is a center, which is always stable. Since sinθ is periodic with period 2π, the critical points (nπ, 0), n=± 2, ± 4, … , are all centers. ② Critical points ± (π, 0), ± (3π, 0), ± (5π, 0), … , Linearization. → (π,0) is a saddle point, which is always unstable. Because of periodicity, the critical points (nπ, 0), n=± 1, ± 3, … , are all saddle points.. (3) Describe the physical meaning the phase plane below. Sol). No damping. i) C.  k;. circular motion. ii). C  k;. circular or semicircular motion. iii). C  k;. reciprocating motion. th. #2. Advanced engineering mathematics, KREYSZIG, 9 , 5.3-15 The associated Legendre functions. Pnk (x)  (1  x2 )k/2. Pnk (x) play a role in quantum physics. They are defined by.  k2  dk Pn 2   and are solution of ODE (1  x )y  2xy  n(n  1)  y  0 . Find  1  x2  dx k . P11(x) and P22(x) . Sol). P(x)  x ⇒ P11(x)  (1  x 2 )1/2 1. dP1  1  x2 dx. 1 dP P2(x)  (3x 2  1) ⇒ P21(x)  (1  x 2 )1/2 2  3x 1  x 2 2 dx. P22(x)  (1  x2 )1/2. #3. Find a solution of. x2y  (x 2 . d2P2  3(1  x2 ) 2 dx. 5 )y  0 . 36. . Sol). Let. y(x)   cm x m  r , m 0. x 2y  (x 2 . (the coefficient of.   5 5  )y  (m  r)(m  r  1)cm x m r   cm x m r 2  cm x m r  0  36 36 m 0 m 0 m 0. x r )=0 ;. 5    r(r  1)   c0  0 36  .  r1  (the coefficient of. x r s )=0;. 5 1 ,r2  6 6. 5   (r  1)r   c1  0 36  . if s=1. (s  r)(s  r  1)cs  cs2  i). r  r1 . 5 cs  0 36. 5 , 6. in (*),. c1  0. in (**),. 2  s  s   cs  cs 2  0 3   c3  c5  c7  ...  0 let s  2p ,. c2p  . 3 c2p2 4 p(3p  1). ( c1 (p.  0).  1,2,... ) 2. 3c 3 c2 c0 3 , c2   0 , c4     4 4 4 2  7  4  2!4  7 2. 3 c4 c0 3 c6      4 3  10  4  3!4  7  10. if s=2, 3, …. (*). (**). p. 5  x 2p5/6 3 9 4 3    y1(x)  c0 ( 1)p    c0x 6  1  x 2  x  ... 896  4  p!1  4  ...  (3p  1)  16  p 0. ii). r  r2 . 1 , 6. in (*),. c1*  0. in (**),. 2  s  s   cs*  cs*2  0 3   c3*  c5*  c7*  ...  0 let s  2p ,. c. * 2p. *. ( c1. * 3 c2p 2  4 p(3p  1). (p.  0).  1,2,... ). p. 1 x 2p1/6 3 3  * 6   y2(x)  c  c ( 1)    c0 x  1  x 2  ...  4  p!2  5  ...  (3p  1)  8  p 1 * 0. * 0. . p. 5 6. 1 3 2 9 4 3    * 6   y(x)  y1(x)  y2(x)  c 0x  1  x  x  ...   c 0x  1  x 2  ...  896  16   8 . th #4. Advanced engineering mathematics, KREYSZIG, 9 , 5.5 Bessel’s Equation. Bessel Function J (x). Derive the solution of the first kind of order ν of x y  xy  x y   y . 2. 2. 2. ( 1)m x 2m 2m  n m!(n  m)! m 0 2 . Sol). J n (x)  x n . th. #5. Advanced engineering mathematics, KREYSZIG, 9 , 6.5-Ex.4 Find y at a certain time t when the driving force is acting as A sin( k / m)t in an undamped mass-spring system, where the spring constant is k and the mass of the body attached to the spring is m. Sol). y  02y  A sin 0 t where 02  k / m .. y(t) . A0  sin 0 t  A t cos 0 t   0 t cos 0t  sin 0t   2  20  0  202.