OT,
DE'THI I: MON PHUONC PHAP TOAN LV
-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)
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
Laplace: y + 3y =
"6 sin .,6 ( t.
utit )
J
Y"
Y.
N$yn* q "A
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-,/