Download Solving Initial Value Problems of Differential Equations and more Exams Differential Equations in PDF only on Docsity!
NIATH 246.
\A\IE:
Exam 1
I I
1) (2 pts) Solve explicitlv the irritial value problern
t:',.1' +
lJ
: r,. AQ)
: (^) 2.
it
V,l
,{r +rl
1"
\
a.
\ i
i^ t
)- (^) )'* )
/
.I
K
L,
:t*rl
e';
I
,t
{^r<
L-
,l
,h't'-;
on
\ ,)/-
- t (^4) -f al -/ l^ L../
z"n
1, tl (^) - / r
X
,r ,
Ylr',t (^) "rl' t l rj^1
t )
Y
r {_^.
,'
-x,
c(.
111,,'f
| (r:^ r*
J"-.a
..,i- (^) {o,.,
\ (^) - r(^
."
.1./ , -.' r.
f.i
l'
,
)a. (^) r c- ld^ 4*^ v.' i.f,...t.t (^) la.-a \
t'(.i
r\ (^) lri
. .,../i^ [l
t'. t"
tL | ,/o* (^) <t*.,. ,1 .,- I
(i.
,/ .q. :i '^ ,' , rJ
,
" -^ 'f
l
e- (^) '1c' r^ Lvt' (^) 'f i-+
\c +'-{ \t"..,.'l..n,.t\
I
/l
L
-. 1 i"I
2) Consider the initial lalue problem
(2r (^) + za)** 10:- (^) + 2y
: (^) 1. ' "' d't'
a. (2^ pts)^ Find the solution in explicit form.
b. (1^ pt)^ Determine the interval of existence.
,lM-"t,
'
"7'
?-V't a+ :) { ?*,';l',i:i^ :.i^
C
.\
'l'r I
-t
(7"t. (^) z-,) .,t
1:,t.
'-
*1"{ tn
{,".d L
Scl,'J,,r^
*i,."' (^) .:
T /i.,f^
|t (^) - ,=
I
17t+?,q,-4,-L:l
t il
l'l
'u/'
i.- l (,
I
(. ".
1 ta\
--
t
i>*t'
'blg sol'*''n'
L
I
'tA
ltrct* (^) u' 4 +(4'
tl
z '-
4a , Ar U ^
i b
7l
u1* ,l^
1b';',
L-
=/
tf
-U
'r,
Y ln \l
n7 , cy')
,t / <-)
tlrl ,ri'^
qr.( \ lvJ:
it:i(t /,rJ (^) ,l' (^) Ll ')a, t€
t,-4 i/. ;11'fd,i< *t'
..S,e''.( (^) t !
(.,\1,'-(-,
f i.^
citl ; ,.^ -^ .,{,(^ tt \
l) (2^ pts)^ -\ tank inititrllv c'onttrins 60 gal^ of^ pure^ \'ater.^ Assume^ that^ n'ater containing^1 lb^
f' '';^
t
of salt/gal enters the tank at a rale of 2 gal/nrin.^ zrnd^ that^ the rvell-stirred^ rnixture^ lear-es
the tank at the rate of 3 gal/min: thus the tank is empt)- after exactll' one hour.
Set up an^ initial^ lalue^ prol-rlern^ n'hich^ s'ill^ enable^ lou^ to^ calculate^ the^ aniount^ of salt^ in
the tank at anv time ur!!l it is emptl'. Do not solve!
,
l,t'. t k^ di..... ,
(.....-, ,-r i
--r
A,t'.
-t
k di..... (^) , L....-,^
,-r i
I C.
t
0 '1,.^. l"rlil
f\ /n', (^) - n txtu) -u
';,,#-- alu rA -^ iJr-r,,^* = (\ lu/t,t\i(24il1^.) - (ft(rtJA^)
I
( {,.-i '-^ t/
(,^ *-^
?-qlf
^,
'r'
Co^t = a
q,Jl^'"
{
tf
,i, y (^) l,
I
3o
/
(L"i"'f
'' (^) /A
V"
L
d
D
'\ v-'-'c
)e.
] (r (^) cr(; A
t ,l
i ,r'*i6l^ valt€,;
,.r
(i .r-(T
'
r-- ..^
1
!{"2 ' o-
,...i
fi
i\
E\ ,
I 'h^
I
A4 A U -'^ 'rX I --r
lt
rc":
, .J;('.! r-"'l ,::+ttit
I
l" ,^ t^ -l(^ ,.'^ c.-^ I t ,,, i
.,'*
rii (^) r1ch.4 r ./ (^) t.-'..'-}(', (^).
"
A rrX tl € ro,,j
'-
@*(
At
L. i,u..
lCrt
{',..,Lr./ t
.t
att '
J..
(=r,
a \X
*,
{t
;i ,r'/,.'^ ,,
-)
.{
:L/-- ,i?/;,
'-P
G(-' f
l\ t
qk r'-trif !-
fr-
Q
('g-'r+'. (^) f "'.
i,"
e4 t'irI
t i1^ lb to,*.
ILr \il. Y,
a
G
frttj \
Running the follori'ing \latlab code has producecl the clircction field below.
[x,Y] (^) = meshgrid(-3. :.2:+3.,^ -2.:.3:+4.);
f'= (^) Q,.^2 + 3.X +^ 1)./(2.V (^) - Z.);
L=sqrt(1. +F.^2)i^ ,^.,?
quiver(X, Y, I./L, F.lL, .5)
(:1'L
1"ct\ =
a. (i^ pt)^ To rvhat clif{crential c<luatiou clocs this clircr:tion field c'on'csponrl? Docs it fit
in er prirticular categorl' of cliffcrential eclutrtions?
yL yl^ ^,7 *'r
-*-
v
t- Ll
r:;ti",^"*r
n'"
(t/,.
(!
b. (1 (^) Irt) Skctch solutions of the diffcrential ecluatiorr fcrund in (a) for initial conclitions
)'(-2):2, )'(2):2 arrcl^ 1'(-3):-
(on the strnre figure that alrezirll'lias thc direction field)
c. (1 pt) \-hat can vou expect of the inititrl lalue
1>roblern corrsisting^ of^ thc^ cliffcrcntial
etluation found in (a)^ ri'ith the initial conclitiorr (^) l'(0):t't \-hr'? If rclo'arrt, add sketches to
the figure.
..'
t'l,.,-',
]l
L i 1
'{'-+'Y
, ?-^ '/. ''^ ( -" -r''/'r
'
?,t (^) q l^ 'i.
, t- r '',, l' (^) ' l'l,.r,^ 4*
isrt'{:4/t€+
c
{(7^,"1a,r.
't-s..av\tae, ,,n (^) €q"C,:t;
( Q1-') Cc'lrro^ e4'*\
1lt
u;E6n*t('.
{ .r I t. { It )' t^
{t -<^ I / (^) )
'l
,J
L+l
t
i L,
:A
l" 1".{
{1. t
: | .')
" /t'\'.^ sr-^ ^ ,
Y
r
,\
i-'"-,
I.
^i / \
\
1r,,
;. tt)
(-r,
I
L-
(1t.-
*l, (^). J
,2.r 2
zz./ / / / / / t tll (^) t I
//////lllllll
----\\\-......-/r'//
/ / / | | | 4l t I t
'+s\\\\a--rr',z,t
/ / / | | | | I
t.l t^ I
l--R-y\rr----u.' / | | | | | t I^ lt
I t I
/_\\r\r\_//,/ | | | t l t t | |
I t I I
/-:\t,,\t:-// | | | t I t I t | |
I | | |
'//
/ / | / / / / /--\
\ \ \ | | | r I t | | | | | |
--//./,//,//2'--\\ \ \ | | | | | t | | t I
--2r'/r'///2?2+\\\ \ \ \ \ t (^) | | | (^) t I /,_\\\\\llllt
i->\\\\\
\\\\\\
\\\\\
\\\\
i
c,..
SQ
{, (^ / 1,"^ ,fi.,
