8/13/2019 MA2001N Week 2 Notes
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MA2001N Differential Equations
Lecture Notes for Week 2
[3. 2nd order, linear odes, reducile
to 1st order for!"
[#. 2nd order odes $it% constant
coefficients"
3. 2nd order, linear odes, reducible to 1st order form
3.1 The general solution
&onsider t%e 2nd order, linear ode of t%e follo$in' s(ecial for!)
*+*+ xfyxqy =
+
. +1*
Note) t%ere is no ter! in y (resent in +1*.
Equation +1* can e sol-ed after !akin' t%e sustitution)
vy = , +1*
$%ere v is si!(l/ a function of x re(resentin' t%e deri-ati-e.
Differentiatin' +1* 'i-es
vy = . +1*
ustitutin' +1* and +1* into +1* 'i-es
*+*+ xfvxqv =+ . +1*
Equation +1* is a first order, linear ode, in t%e -ariale v , and can e sol-edfor v , / usin' t%e inte'ratin' factor !et%od.
a-in' found v , t%e de(endent -ariale,y
, follo$s fro! +1* / carr/in' outanot%er inte'ration. After doin' suc% an inte'ration, $e $ould %a-e)
+= Adxvy ,
$%ere A is a constant.
o far, $e %a-e outlined t%e 'eneral idea e%ind 4reducin' t%e order of an equation
of t%e for! of +1*. A $orked e5a!(le is no$ needed to clarif/ t%e a((roac%.
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8/13/2019 MA2001N Week 2 Notes
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MA2001N) Lecture Notes for Week 2 +continued*
3.1 6%e 'eneral solution +continued*
Worked Example: reducible e!uation
7ind t%e 'eneral solution of
122 =+ yxyx , 1x . +18*
999999999999999999999999
:sin' t%e sustitution vy = , fro! +1*, and follo$in' t%e !et%od
outlined ao-e, equation +18* eco!es
2
12
x
v
x
v =+ . +20*
Equation +20* can e sol-ed / t%e inte'ratin' factor !et%od. o,
"ln2e5(["2
e5([ xdxx
I == and
2xI= .
After !ulti(l/in' t%rou'% / I , equation +20* eco!es
122 =+ vxvx
and
1*+ 2 =vxdx
d .
;nte'ratin' 'i-es
1
2 cxvx += ,
$%ere 1c is a constant, and it follo$s t%at
2
11
x
c
xv += .
8/13/2019 MA2001N Week 2 Notes
3/13
MA2001N) Lecture Notes for Week 2 +continued*
3.1 6%e 'eneral solution +continued*
Worked E5a!(le) A reducile equation +continued*
;nte'ratin' +21* no$ 'i-es
21ln cx
cxy += +22*
$%ere 2c is a second constant.
999999999999999999999999999999999999999999999999999
6%at concludes t%e $orked e5a!(le. Notice %o$ t$o constants %a-e
a((eared in +22*. No$ tr/ E5a!(le %eet 2+a*+i*.
3.2 "nitial conditions and initial #alue problems
6%e 'eneral solution of a 2nd order ode $ill al$a/s contain t$o aritrar/
constants. 6%e $orked e5a!(le in section 3.1 %as 'i-en a de!onstration
of t%is, $%ere equation +22* contains t%e constants 1c and 2c . 6%ese t$o
constants can e e-aluated, (ro-ided suitale conditions are 'i-en.
As a co!!on e5a!(le of suc% conditions, $e introduce $%at are called
initial conditions. 6%ese 4conditions are referred to as 4initial, ecause
t%e/ are 'i-en at t%e start of a (rocess. 6%e/ are 'i-en in t%e for!)
0yy= at 0xx=
and
0yy = at 0xx= ,
$%ere 0y , 0y and 0x are 'i-en constants.
An alternati-e $a/ to $rite t%ese conditions is
00 *+ yxy = and 00 *+ yxy = .
An initial -alue (role! is one in $%ic% a 'i-en ode %as to e sol-ed
to'et%er $it% t%e a((ro(riate nu!er of 'i-en initial conditions.
6%e $orked e5a!(le in section 3.1 can no$ e e5tended to for! an
initial -alue (role!, t%us s%o$in' %o$ suc% initial conditions can e
used to e-aluate t%e aritrar/ constants in +22*.
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8/13/2019 MA2001N Week 2 Notes
4/13
MA2001N) Lecture Notes for Week 2 +continued*
3.2 ;nitial conditions and initial -alue (role!s +continued*
Worked Example: n initial #alue problem
ol-e t%e follo$in' initial -alue (role!)
122 =+ yxyx , 1x ,
su=ect to t%e t$o conditions
0=y at 1=x and 0=y at 1=x .
7or succinctness, t%is initial -alue (role! $ould e stated as)
122 =+ yxyx , 0*1+ =y , 0*1+ =y , 1x . +23*
999999999999999999999999999999999999999999999999999
We %a-e alread/ found t%e 'eneral solution of t%e ode in +23*. ;n
su!!ar/, fro! +22* and +21*, $e %a-e
+22*) 21ln cx
cxy += > +21*)
2
11
x
c
xy +=
No$ usin' t%e 'i-en conditions fro! +23*, na!el/
0*1+ =y and 0*1+ =y ,
in con=unction $it% +22* and +21* res(ecti-el/, $e %a-e
t$o si!ultaneous equations)
21*1ln+0 cc += and 110 c+= .
7ro! t%ese, t%e t$o constants can e found as)
11 =c and 112 == cc .
o, fro! +22*, t%e solution of t%e initial -alue (role! is
11
ln +=x
xy .
999999999999999999999999999999999999999999999999999
6%at concludes t%e $orked e5a!(le. No$ tr/ E5a!(le %eet 2+a*+ii*.
#
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5/13
MA2001N) Lecture Notes for Week 2 +continued*
$. 2nd order odes %ith constant coefficients
;n t%is section +#*, $e $ill consider t%e case $%en t%e 2nd order ode is %o!o'eneous,
t%at is, t%e ri'%t?%and side is @ero. We $ill also e assu!in' t%at t%e coefficients ont%e left?%and side of t%e 'eneral linear for!, 'i-en in +2*, are all constants.
o, let t%e 2nd order, %o!o'eneous ode $it% constant coefficients e)
02 =++ ycDybyDa , +2#*
$%ere a , b and c are 'i-en constants.
A solution, y , is no$ required, $%ic% satisfies equation +2#*.
8/13/2019 MA2001N Week 2 Notes
6/13
MA2001N) Lecture Notes for Week 2 +continued*
#. 2nd order odes $it% constant coefficients +continued*
Equation +2* is, of course, a quadratic equation and so it can e sol-ed for
as)
a
acbb
2
#,
2
21
= , +28*
$%ere 21 , are t%e t$o roots of equation +28*, de(endin' on $%ic% si'n res(ecti-el/ is c%osen.
As usual $it% quadratic equations, t%ere are t%ree (ossile cases to consider)
&ase 1) oots of +28*, 1 , 2 , are real and distinct>
&ase 2) oots of +28*, 1 , 2 , are ot% co!(le5>
&ase 3) oots of +28*, 1 , 2 , are real and equal.
Eac% of t%ese cases $ill no$ e dealt $it% in turn.
$.1 &ase 1: 'oots of (2)*: real and distinct
W%en acb #2 > , equation +28* $ill 'i-e real and distinct -alues for 1 and
2 . Equation +2* t%en 'i-es t%e t$o solutions
xeky 111
=
andx
eky 222
=
corres(ondin' to t%e t$o roots 21 , . 6%e 'eneral solution of +2#* is t%en
t%e su! of 1y and 2y , $%ic% 'i-es
xxekeky 21 21
+= , +30*
$it% 21 , fro! +28*.
$.2 &ase 2: 'oots of (2)*: both complex
W%en acb #2 < , equation +28* $ill 'i-e co!(le5 nu!er -alues for 1 and
2 . 6o de-elo( t%is furt%er, $rite equation +28* as follo$s)
+continues o-erleaf*
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7/13
MA2001N) Lecture Notes for Week 2 +continued*
#.2 &ase 2) oots of +28*) ot% co!(le5 +continued*
a
bac
ia
b
2
#
2
2
1
+=
and
a
baci
a
b
2
#
2
2
2
= ,
$%ere i is t%e unit i!a'inar/ nu!er, 1=i , and $%ere it s%ould e
noticed t%at, / re-ersin' si'ns, t%e ter! 2# bac is no$ real. We also
take t%e ter! to e strictl/ (ositi-e.
6%e roots, 21 , , !a/ e are-iated as
i+=1 , i=2 , +31*
$%ere
a
b
2= ,
a
bac
2
# 2= , +32*
6%e 'eneral solution can no$ e found / co!inin' +30*, $%ic% still a((lies
e-en t%ou'% t%e roots are no$ co!(le5, and +31*. 6%is 'i-es)
xixi ekeky *+2*+1 +
+= . +33*
6akin' out t%e co!!on factor, xe , +33* eco!es
*+ 21xixix ekekey += . +3#*
;t is i!(ortant to realise t%at t%e solution, y , as 'i-en in +3#*, is a co!(le5 nu!er. We are, %o$e-er, lookin' for a real solution. 6o find suc% a solution, $e need to
de-elo( t%e ter!s in +3#*, usin' t%e t%eor/ of co!(le5 nu!ers.
After usin' t%e t%eor/ of co!(le5 nu!ers, it can e s%o$n t%at +3#* can e $ritten)
*sincos+ 21 xkxkey x += , +3*
$%ere 1k and 2k are no$ real -alued constants and and %a-e een defined
in +32*. Note t%at in +3*, y is no$ a real -alued solution, $%ic% is $%at $e are seekin'.
8/13/2019 MA2001N Week 2 Notes
8/13
MA2001N) Lecture Notes for Week 2 +continued*
#.2 &ase 2) oots of +28*) ot% co!(le5 +continued*
n aside: +o% does e!uation (3$* become (3*-
;n equation +3#*, y , 1k and 2k are, in fact, all co!(le5) y is t%e
co!(le5 solution, sa/, 21 yiy + , $%ereas 1k and 2k are co!(le5
constants. ;t is t%erefore !ore a((ro(riate to $rite +3#* as)
**+*++ 221121xixix
ebiaebiaeyiy
+++=+ .
$%ere 1a , 1b , 2a and 2b are real constants.
8/13/2019 MA2001N Week 2 Notes
9/13
MA2001N) Lecture Notes for Week 2 +continued*
$.3 &ase 3: 'oots of (2)*: real and e!ual
W%en acb #2 = , equation +28* $ill 'i-e real and equal -alues for 1 and
2 . ;t follo$s fro! equation +28* t%at 1 + 2= * is 'i-en /)
a
b
21 = . +3*
ince 1 is no$ t%e one and onl/ root of +28*, it follo$s t%at equation +2*
$ill 'i-e one +and onl/ one* solution, $%ic% $ill e)
xeky 111
= , +3*
$%ere *2C+1 ab= %as een 'i-en in +3*.
8/13/2019 MA2001N Week 2 Notes
10/13
MA2001N) Lecture Notes for Week 2 +continued*
#.3 &ase 3) oots of +28*) real and equal +continued*
n aside: +o% do %e deri#e e!uation ($* from (3)*-
We !ust sustitute 2y fro! +38* into t%e equation $e are tr/in' to
sol-e, $%ic% is +2#*. 6o do t%is, $e need to find t%e deri-ati-es 2y
and 2y . o, fro! +38*, after differentiatin' -arious (roducts, $e %a-e
112 yvyvy += +#3*
and
1112 2 yvyvyvy ++= . +##*
ustitutin' +##*, +#3* and +38* into +2#* no$ 'i-es
*2+ 111 yvyvyva ++
*+ 11 yvyvb ++
*+ 1yvc+ 0= .
No$ collect to'et%er all ter!s in v , in v and in v , to 'i-e
*+ 111 ycybyav ++
*2+ 11 ybyav ++
0*+ 1 =+
yav . +#*
&onsiderin' t%e v ter! in +#*, $e kno$ t%at 1y is a solution of+2#*.
o, i!!ediatel/ $e kno$ t%at 1y can re(lace y in +2#* to 'i-e
0111 =++ ycybya . +#*
Alread/ it can e seen t%at +#* is 'reatl/ si!(lified.
No$ considerin' t%e v ter! in +#*, $e can e-aluate t%e factor
*2+ 11 ybya + as follo$s. ince, fro! +3*,xeky 1111
= , $%ere,
fro! +3*, *2C+1 ab= , it follo$s t%at
0**2
+2+2 1111 =+=+ ba
baekybya
x+#*
Equations +#* and +#* s%o$ t%at, in +#*, ot% t%e ter! in v and t%eter! in v are @ero. Equation +#* t%erefore reduces to t%e -er/
si!(le
for!)
0*+ 1 = yav .
8/13/2019 MA2001N Week 2 Notes
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8/13/2019 MA2001N Week 2 Notes
12/13
MA2001N) Lecture Notes for Week 2 +continued*
#.3 &ase 3) oots of +28*) real and equal +continued*
n aside (continued*
7or non?tri-ial solutions, 01 y , $e t%erefore %a-e
0=v .
;nte'ratin' t$ice no$ 'i-es
xccv 21 += ,
s%o$in' t%at, in t%is case, v !ust indeed e a 'eneral linear function,
as %as alread/ een su''ested in equation +#0*.
End of the aside: We ha#e no% deri#ed e!uation ($* from (3)*-
6%e a((roac% used %ere to deri-e +#0* and t%en +#1*, t%at is, to find a second
solution, 2y , $%en a first solution, 1y , is kno$n, is a (articular e5a!(le of a
(o$erful 'eneral !et%od called t%e !et%od of eduction of rder. 6%is
$ill e studied furt%er in section .
$.$ &oncluding comment in summar/ of section $
;n (ractice, $%en doin' (role!s in 2nd order odes $it% constant coefficients,
it s%ould e realised t%at onl/ -er/ fe$ ste(s need to e $ritten do$n.
Note, in (articular)
&ase 2) %a-in' found and , equation +3* is used directl/>
&ase 3) %a-in' found 1 , equation +#2* is used directl/.
6%e (rocess is est illustrated / $orked e5a!(le, as follo$s.
Worked Example: 2nd order odes %ith constant coefficients
ol-e t%e 2nd order ode
0#2 =++ ycyDyD , +#*
for t%e t%ree cases 3=c , =c and #=c .
9999999999999999999999999999999999999999999999999999
+continues o-erleaf*
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8/13/2019 MA2001N Week 2 Notes
13/13
MA2001N) Lecture Notes for Week 2 +continued*
&ase 1: 3=c +6%is $ill 'enerate t%e case in section #.1.*
7ro! +#*, $rite do$n t%e au5iliar/ equation directl/.
+&o!(are +2#* and +2* in t%e 'eneral deri-ation.* o, in t%is case)03#2 =++ .
ol-e t%is to 'i-e t$o distinct real roots
31 = , 12 = .
7or t%is case, equation +30* a((lies. o, directl/ fro! +30*, $e %a-e
xx ekeky += 23
1 .
&ase 2: =c +6%is $ill 'enerate t%e case in section #.2.*
7ro! +#*, $rite do$n t%e au5iliar/ equation directl/ as)
0#2 =++ . +#8*
ol-in' +#8* 'i-es t%e co!(le5 roots
2C*#1.#+, 21 = ,
or
i+= 21 , i= 22 .
6%ese are in t%e for! of +31* and so and can e identified as
2= , 1= .
7or t%is case, equation +3* a((lies. o, directl/ fro! +3*, $e %a-e
*sincos+ 212
xkxkey x
+=
.
&ase 3: #=c +6%is $ill 'enerate t%e case in section #.3.*
7ro! +#*, $rite do$n t%e au5iliar/ equation directl/ as)
0##2 =++ .
6%is equation %as t$o equal roots, 21 = , and after sol-in' it
21 = .
7or t%is case, equation +#2* a((lies. o, directl/ fro! +#2*, $e %a-e
xexkky 221 *+
+= .
6%at concludes t%e $orked e5a!(le. No$ tr/ E5a!(le %eet 2+*.
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