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Homework 2 (IntegratingFactors,SeparationofVariables,andExistence&Uniqueness) MATH 214 - Spring 2025
Instructor: Joanna Ro Name:^ SQ~~~~
Unless otherwisestated,pleasewriteall^ solutionsinexplicit^ form.
9(4^ points^ each)^ Without solvingthedifferential equation, determinethelongest interval on which a unique solutionto thegiven IVP is guaranteed^ to^ exist.
llt(t - 3)x' + (t 2 - 9)x =0, x(l)=-
~;t:>~ ~ 0
x' +- l t -t3')x. ~ 0 --------^ .,,
?lt) ~lt)
? °'~ ~^ (!Jr()__^ lM"V.v\Jv'-)^ ~..r ~\^ t^ t-l-rlJ /oO).
--- \ \r\v ., +V\J... \ \J ~ ~~ ~ _wrOf_ ~uJ -to ~ <>-- v. _'I_ r~w. s ~ \ V-1\ ~
~ CA\ -tt-l-oe,oi::>).
2
-(e^2 t - l)x'+ t 2 + 4 x =^ ln(t - 2), x(5)^ =-
x'
'L 't'^ l>t---"L~ -t- l e,1.,t- ^ ')^ l^ t'..^ ~^ Lf)^ '/..
:. t,'U;^ -\
........__ ""-.r _..,
~
? l~) l) l~')
-c~O ,t:f.O
-t~1..
v d^ WA~1/\JvN-.j f!Y'. -oC,0) ) ..0, oO)
~
~J (^) C Y, t--,.1/\ -..A~J O,A. l1-,o0')
s~ Q ~, ~ s,^ f).._^ V.V_^ :q"""^ ~,^ v,.'t;-,Y.^ --0^ ~^ ,^ \J~ ;^ ~ 5
V,.l,\VW'\ -{..t c>-
~ {.:~. ~ ~ fJ~v 1Ct; ( 0 , c,0)^ _(_^ ^ 'L^ ,~) -=- (^ L^ )^ oO^ )
1' \f~St~
(
a 1tx' +tsin(4t)x = e-t, x(-1) = 0
' \ , I '\
'f.... -t-1-- ..)\f_ \ 4..::. )x
\
pl~
~ '~ _L~t\V\/\M) ~.. l-oO, ~) ' l ()' oe')
~ _,V_ lL "t-v-::.- \ , TN-- \ V? ~ (l v\l,'l"VI_ >ru.J ,0 **V\f/\1( tr-** v. If\ ~ """- s.,) ',,·,
1"1¥ ~\ to-l-~,oi:l)(~-o0,DJ =- l-~,J)
.... ...tx I (^) --x=tan.t, 2 t-
x(l) = 2
\ "L
~ - t lt-S) X -
511/\ \ t')
-twilt') ~
C:1 .:t'J
'----.....-----
p L+-')
t~ 0, S (^) -t ~ Vt"'-r, 1 't, , _(_ 1c 7L
S ~l.9- -t :> =- \ t- l D,S) n \ O, 11.~'\ =- ~ 0, ~)) -t'A~ (^) ,~ -hL
\Y\ -(p/Vr,..\ .y,._ v-.N,Vf_ e,.. V\V'.·, r,, vJ- ~0\v,'1-Y' -") -IN_ '-'? w.
d\A (J,.vOA -Uffi- h {, )(, ~ ~-
-JIA..^ Solve^ the^ initial value problem^ for^ the initial condition^ x^0 =l. Are there any restrictionson^ t^ for
existence of solutions?
- )<i. ~ ~ \V\^ \ ~-:1 ^ -- (_
xlt')::. +-Jc-~'""^ \t-~~\
..Solve the initial value problem^ for^ the initial condition^ xo^ =^ 0.Does a solution exist? Isitunique?
0 :: ,x:l0')^ =--^ ~^ JG^ =-')^ L^ =-^0
xlt-Jcc_+^ J-:),/\\Hl\
~~ \v,--.V!,^ Vi,^ &^ T^ ~^ -^ '.:>"'\ ^ -rt~ ^ '>^0 1 \oJ,/r^ 'rW.,'
wf"Q_ '{_ c.)-- Vv'I\ ~ qw___.
- (5points each) Solvethefollowing differential equations usingthemethodofintegrating factors.
(a) x' - Bx= 12e^8 t
-Rt
~
t - t¾" ~ =-^ j \1...^ d.-:"^ ==-^ VL^ ~^ -- C
K lt) -:. e.,^ tt^ l\1...t^ ~^ L^ )
dx 2
(b)2- dt -^ -xt^ =^ t
i(-t-1")^ =-^ ~
-:-)( ~ J-t clt
■(5 points) Solve the followingIVPusing separation^ of^ variables. You may leave your answer^ in^ implicit form.
dx 2t3 + 1
dt = 5 - x 4 '
j l S - x "')^ c\i<. =- \ (^ l.-:^3 +"I)^ cH;
s~- ix.s +(_
x(O) = 1
6. (5 points each) Solve the following IVPs using^ the^ method of integrating factors.^ Describe the limiting behavior of the solutions.
(a) tx'^ +^ x^ =^ 3tcos(2t) ,^ x^ (;)=^!
x' -t-^ t^ X^ -=-^3 lDJ^ ~^ 1.--t)
tx ~^ J3-t-coJ (^ 1.,-)~t e,\A..-sa..t
t )(_:^ t^ ts,V'(^ 1k)^ -^!^ ~^ s^ ~^ l't~)o-t
=- ~~~,~ \tk) ~^1 '-\LQ^ ~vt') ~<-
) :. t ~ \Y" ~ '2..,-t'~ c)..v ::.. t.OJl1---t > d~
-') ~ Ll) -=- -t5""'^ l1,,t')^ ~^ i tOJlV) +- t
j_=-x.(tJ^ ~^ -l-. -^ 'Z..
11 .,\ ii
')(.l ~ "):::. ;^ ~^ rnl'L~')
(b) tx' =-(t+^ l)x^ +^ 4te-t^ ,
~ 1.<..^ -^ --^3
~ ¼t C.,DJ\1., -')
5 x(l) = - e
+-^ LlL^ L.<...-
::. 'L1' l..,Ti
s -+- -Uvt
J~-"""t-Jc>-t- t +-l~-t /'L-l') = -e..^ ~^ e.. :::-^ tet
;-<.¾ x ::. 1 L\tot^ ~^ 1-.-ct.^ ~^ c
=> X Cir) ~ '2..t e --t .- c.... -:_,~'t :.^ e..^ - -1: (^ 1,t .--^ ~^ J
½ =-X l I J ~ e-_ \ ( 1- +^ l.J
s
x lt- ') ~ ~- t (^ 11:^ ~^ t^ )
)( l~) ~ Q^ £kJ *~^ c;()
It(5points each) Solve^ the^ following differential equations using separation of variables.
dx
•- dt =tan(x)
\V"^ ^ ~~~^ t~'\ ^ =-^ "t^ +-^ L
~"""(;<.) :. CI~"G
~lt')-:.^ ~-,1/.., l~t-\J)
x 2 -x-
•x'=--- t
) .A-,-~^ ~^0
l^ L-\A^ -<;~^ ~^ \
~ JL~~s -,_~)ch -^ \v-,^ \1.-\ ..- c
\ ~ \ :-.-:\ -^ >'\^ -t^ _0_^ ^ 4-- (.,
x.-^ S^ : (^ t^ ~^ l^ "'~l-)^ ==-^ C...t
(^4) )( ~ 4 L-t"
( \ - Ct~) x :-^ L^ C-t^ °'^ ~^ S
~lt-'): 4 Ct" +-^5
\ - Ck:~
..(10 points) Considertheinitial value problem
x'^ =^ 4x^ +^ 2e^2 t^ -^ 1 ,^ x(O)^ =^ xo^.
-Solvethe^ IVP. (Your answer should^ be^ in terms of^ x^0 .) x' - l.\x =- 1.e.i.,t-^ I^ ;VU-)^ =-^ eJ-^ L,;^ t^ ~^ e.,-^ l..lt
e-'-'•<^ =-^ f('u..-"'--e.•U-t) Jt
~ -;. xlo")^ - - \ -r ¾^ +-^ L-
\ ~ t ~~ ~ i; ) e, .lt-
a Describe how the^ long-term behavior of^ the^ solution depends on^ the^ value of^ x^ 0.^ (Hint: There is some critical valuex8ritsuchthatsolutions grow negatively or positively depending^ on^ whether^ x^0 >^ xgrit or x8"t· Don't forget^ to^ consider^ what^ happens when^ x^0 =^ xgrit.)
~3--\.,...^ ~^ ~v..:r^ i9~./v~Y.^ c,.'-:,~^ ~~^ o,...J^ ~~^ rP^^1 ~^ ^ ~4^ ~^ ~)^ l^ '{~
~ vJ..\ ~,v,.A-(..
\t- 'f-.o ( - ~^ '3> ) ~ ~l-"')^ ➔^ -o0.
~^ -,.,, '> -
~ -l..^ ) ~^ X.^ l+-)^ ➔^ o0^ I
~ 1'0 =- - (^3) - ~ X\l() ➔^ -oO ' '--\ '
&lr.(10 points) Find a general solution^ ofthefollowing differential equation.^ Then, describe howthelong-
termbehavior ofthesolution depends on thevalueof^ x 0 =x(O}. (Hint:^ Can^ you analyzethe^ DEtosee
how solutions behave?)
x' - x 2 = -
~^ - -^ -^ - \1..
,~\ )(- ^ ^ ~^1 ~ ~^ (._, X-r\
'I.. - \ ::. ( {, 1.,- l X. or-) -^ C..,e.,^ LtX T"""^ Ce,^ 1.,t
'x.( \ - lL·u-) ;:,_ Ce.- 1.,t .~ \
(.,e,'lk +-
x l~) -:. I - (_,e,., u
\t- )(D ~ \ _rV.VI_ xtY)-=-
l +.>r CA,\ tbl-o0,...o'). \t- (^) *'I...~-=--* ~/\ )( ( t-'~ ~ -
-w^ (X\^ 'ttTl-~,c,t)').
+- Xo <.- \ t'Y..UI\ xl~,-=,,,-\ 0...1 t-....., oO ' \t' - ^ (.^ Xo^ '-. ^ I ~^ x.^ lt-)^ ~^ - ^ (>,.J t~o
ll^ ~o^ ') ^ _t-V\1,\1_^ x.Lt-')~^ ~
Q...j t~ 0().
