4 OT, DE'THI I: MON PHUONC PHAP TOAN LV i, (Thoi gian: 120 phrit l1len MQt)- O( - Cdu I: (3 di€m) L Tim dao dQng tr,r do u(x, t) c0a soi ddy ddi vd h4n vdi rti€u ki€n ban dAu: luQ, t) = -f (x); vx e R I#r,=,- r(x); vx e R I 2. v€ hinh d4ng ddy t4i thoi di6m r, = - (a ld v4n t6c truy6n s6ng) vdi itiAu kign [o khi x <o;x > 2 I F(x) = 0 vd /(x) = Jx khi 0. -< x <l lz-, khit<x<2. khdng chria nguiin nhi€t thda Cdu II: (3 diem) . l. Tim nhipt d0 u(x, t) tr€n thanh din nhi€t c6 d9 dei / r.r luQ, t) = u(1,/) = 0; Vr ) 0 oreu Kten: < '-''"' Lr(r, 0) =./(x) ; Vx. [0, /] 2. Tim nhiQt dQ u(x, t) th6a phuong trinh truydn nhiQt: ^ ^) Ou -d'u x ()-r- at -' ox' ' et \ \ Lt ? r:l , !^ fu(O, t) = u(1,/) = 0; Vr > 0 tnoa oteu Kten: < ,/_ * *- _ -*"=y: r,"- ^,vu._|1p60) =.0 I V" e [fi li '*\ V{(t - a)H(t t<a :' t>o a)) = e-u'F(s) LJ"L F(s) thi Io: "l = lr,' vE: Cdu III: (2 <li€m) l. Chung minh ring: Ni5u Y{f(t)} : Trong tt6 hdm Heaviside H(t - 2 Cho hdm sO f(t) c6 dd th! nhu hinh - f(r\ Laplace: y + 3y = "6 sin .,6 t. ( utit ) J Y" Y. N$yn* "A q a) Dtrng hdm Heavisde it6 tim bieu thuc c0a him f(t). b) Tim bir5n OOi Laplace Y{f(0} Cdu lV: (2 ttiCm) l. Cho him g6c f(t) = t.cosrt . Tim hdm inh F(s): ytf(t)) 2. Tim nghiQm ri0ng y = y(t) cta phuong trinh viphdn sau dAy bing ph6p bitin ttdi l?. 14s6 -Tt Vbi y(0) : -l; y'(0) = g nt$4ti-,/ . 4 OT, DE&apos ;THI I: MON PHUONC PHAP TOAN LV i, (Thoi gian: 120 phrit l1len MQt)- O( - Cdu I: (3. e [fi li '* V{(t - a)H(t t<a :' t>o a)) = e-u'F(s) LJ"L F(s) thi Io: "l = lr,' vE: Cdu III: (2 <li€m) l. Chung minh ring: Ni5u Y{f(t)} : Trong