Question

1. Solve the initial value problem for a damped mass-spring system acted upon by a sinusoidal force for some time interval f(t) = {10 sin 2t 0 0<t< y(0) 1, y'(0) -5 y"2y' 2y f(t), Tt zusor= 2. Consider two masses and three springs without no external force. The resulting force balance can be expressed as two second order ODES shown as below. mx=-(k k2)x1+ kzx2 m2x2 (k2k3)x2 + k2x1 15 If m 2,m2 ki = 1,k2 = 3, k3 = 4 a. write these two second order ODES as a system of first order ODES. Your coefficient matrix should become 4 by 4 matrix. b. Find eigenvalues and eigenvectors of the coefficient matrix c. Find a general solution of the homogeneous system 3. Find a series solution in powers of (x - 1) (i.e., xo = 1, # 0) the following equation y" - xy 0,-o <x < oo

Answer 1

+ ay fle) fe)- LO Ain (dt 0t ス<も 0t<5 dy10 Ain(&t) Aalution i th inlution homagonco eguakion and ) barieulas intognal An (at DDt shusatary atin t ant= 0 - V-4alx2 - cs Scanned with CamScanner -

datinat tain cal resla A then C, e TR lacse d, d amd de 1 aiway cenatant naellry guatuica has eal nal Haca Auta is allay d-i taa tes Scanned with CamScanner

Auke 3 ) Now', (10 Ain(dt)) :() Mn pla (krovidd nminator i Non Aim (at) (ain (8t)) x OT (Da+ (3t) Ain (st) Ain (at) 5 X D-) Scanned with CS CamScanner

5 D+1 Ain (at) ngain Jufolae D+ 1 An (at - 4 -L (DtI) Ain (at (5 (D+i) - 2) Ain sinldt) inlat =) ces (at - E) laces (t) Cin e - Aim (dt) (4 e Ain x 0 as Scanne witr CS CamScaner a = L

C, 3 t e (-) (c, ces t C, Aw t) (-1) + e ) ) C Ai) + 4 An aolo (e,+0) - 2. (a -3 t -2 - 3 C-5 as -5 5 -5 CS Scanned with CamScanner

c t)B ain .. A + iolutuim it i ertinveus and ind fria cenditin wang A and 8 we nil A - (Ax) +Bx0 e in 3 (-) 2Xi -0 -se -2 3e -Ae-7 3 teT Scanned with CS Camsner G ampithig ceutd

t ces lt)- Les(at) B CS Scanned with CamScanner