MA2001N Week 2 Notes

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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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 += .


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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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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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$. 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#*.


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#. 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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#.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'.


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#.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.


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$.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*.


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#.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 .


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#.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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&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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MA2001N Week 2 Notes

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