3c DIFFERENTIAL EQUATIONS.doc

8/21/2019 3c DIFFERENTIAL EQUATIONS.doc

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CAPE MATHEMATICSUNIT 2 MODULE 3: COUNTING, MATRICES AND DIFFERENTIAL

EQUATIONS

(C) Differenti! E"#ti$n% n& M$&e!!in'

Students should be able to:

1. solve first order linear differential equations y - ky = f(x) using an integrating factor, given

that k is a real constant or a function of x, and f is a function;

2. solve first order differential equations given boundary conditions;

. solve second order ordinary differential equations !ith constant coefficients of the for"

y

# ay

# by = f(x), !here a, b, c∈

and f(x) is$

(i) a %olyno"ial,

(ii) an ex%onential function,

(iii) a trigono"etric function;

and the co"%le"entary function "ay consist of

(a) 2 real and distinct roots

(b) 2 equal roots

(c) 2co"%lex roots

&. solve second order ordinary differential equations given boundary conditions;

'. use substitution to reduce a second order ordinary differential equation to a suitable for".

Intr$ti$nAn equation that involves an unknown function and one or more of its derivatives iscalled a differential equation.

EXAMPLES:

1. x x f x f =+′ ))

!. "!! !!

!

+−=+− x x ydx

dy

dx

yd where y # f x).

$he stud% of differential equations has two &rinci&al 'oals:

1. to discover the differential equation that descri(es a 'iven situation!. to find the a&&ro&riate solution of that equation.

Solvin' a differential equation involves findin' the unknown function* f * for which thedifferential equation is true on some interval of real num(ers.

Man% natural &henomena involve chan'e. $he derivative of a function ma% (e re'ardedas the rate at which a quantit% is chan'in' with res&ect to the inde&endent varia(le.

+ifferential equations are often used to descri(e the chan'in' universe.

EXAMPLES:1. ,f k is a constant* then ever% function* f * that satisfies the equation

f ′ (t) = k f(t)

has the form f(t) = C e kt

-APE P/E MA$0EMA$,-SModule !.1c) ,nte'ration ,, +ifferential EquationsMM !22" /evised !213

1


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for an a&&ro&riate constant C .

Such a function re&resents e4&onential 'rowth if k 5 2 ande4&onential deca% if k 6 2.

!. e!tons *a! of +ooling$ The time rate of change of the temperature,T(t), of a body is proportional to the difference between T and thetemperature, A, of the surrounding medium* ma% (e modelled (% thedifferential equation:

) , -k dt

d, −=

where k is a &ositive constant.

3. $he motion* v* of a fallin' o(7ect ma% modelled (%t

dt

dv8.9−= .

. $he chan'e in the si;e of a &o&ulation* * ma% (e modelled (%

+ k. dt

d. =+ *

where k and C are constants.


3/17

'iven initial or boundary) conditions* which s&ecif% the value of the solution or certainof its derivatives at s&ecific num(ers in the domain* are satisfied.

Solutions of differential equations can (e e4&ress in the form y = f(x), an explicit solution

or k(x,y) # 2* an implicit solution!

Te Or&er $f Differenti! E"#ti$n$he order of the differential equation is the order of its hi'hest derivative.

EXAMPLE:

+ifferential equations with y′ ordx

dyas the hi'hest derivative are called first order

differential equations

+ifferential equations with y′′ or!

!

dx yd as the hi'hest derivative are called

second order differential equations

Fir%t Or&er Differenti! E"#ti$n%

A. Equations that ma% (e e4&ressed in the form )) x g dx

dy y f = .

$his t%&e of differential equation is solved (% se&aratin' the varia(les and inte'ratin' (oth sides of the equation with res&ect to x* the inde&endent varia(le.

See nit 1 notes.)

E*eri%e 2+3++ Past E4amination =uestions)

1. >ater lilies are s&readin' over the surface of a &ond at a rate which is &ro&ortional to the area alread% covered (% the water lilies. $he area coveredat time t is A(t)* where A is a function* assumed to (e differentia(le. ?(tain*with e4&lanations* a differential equation for A(t), which models this situation.

@ive reasons wh% the descri&tion of the s&read of the lilies (% means of adifferentia(le function is 7ustified.

[Specimen Paper 1 – 6 marks]

!. $he rate of increase of a &o&ulation P is &ro&ortional to P with the constant of &ro&ortionalit% (ein' 2.23. >rite down a differential equation model for P.


3


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,f the initial &o&ulation is P2* find P at time t. [1999 Paper 1 – 3 marks]

3. $he rate of decline of an insect &o&ulation due to the a&&lication of a certaint%&e of insecticide can (e modelled (% means of the differential equation

t dt

dx

A1

A22

+−= *

where x is the num(er of insects alive t hours after the a&&lication of theinsecticide.

,f there were 1222 insects initiall%* calculatea) the num(er of insects alive after ! hours [ marks]() how lon' the &o&ulation of insects will survive. [! marks]

[!""" Paper 1 – 6

marks]

. $he rate at which an air freshener (lock eva&orates is directl% &ro&ortional toits volume. At time* t weeks* the volume of the (lock is " cm3.

i) +enotin' the &ositive constant of &ro&ortionalit% (% k * write down adifferential equation relatin' " and t . [! marks]

ii) 0ence* show that kt -e/ −= * where A is a constant. [marks]

iii) @iven that A and k are (oth &ositive* sketch a 'ra&h showin' the variation

of " with t . [! marks]iv) ,nitiall%* the air freshener (lock has volume cm3. ,t loses half itsvolume after weeks. -alculate the e4act values of k and A. [# marks]

v) $he air freshener (lock (ecomes ineffective when its volume reaches cm3. -alculate the time* to the nearest week* at which the (lock should (e re&laced. [3 marks]

[!""1 Paper ! – 16 marks]


5/17

E4act +ifferential Equations equations that ma% (e e4&ressed in the form))))) x h y g x f y g x f =⋅′+′⋅

Since the left hand side of this equation is the derivative of f x) g y)* i.e. )B)C y g x f dx

d *

the solution of equations of this t%&e is

∫ ∫ = dx x hdx y g x f dx d

)B))C

∴ ∫ = dx x h y g x f )))

EXAMPLE

Dind the 'eneral solution of the differential equation

x e y

dx

dy x =+

0olution

) xydx

d y

dx

dy x =+

∴ x

e ydx

dy x =+

⇒ dx edx xydx

d x ∫ ∫ =)

⇒ ce xy x += where c is the constant of inte'ration.

$%ercise !&3&c&b



6/17

>orksheet E4act +ifferential Equations

' Linear Dirst ?rder +ifferential Equations equations that ma% (e e4&ressed in the

form ) x f kydx

dy=+ * where k is a constant or a function of x.

>e use an integrating factor to solve this t%&e of differential equation.

$he integrating factor is kx dx k dx yof t coefficienthe eee =∫ =∫ )

or ))) x#dx x g dx yof t coefficienthe eee =∫ =∫ where

∫ =

)) x#dx x g

Cote the constant of inte'ration is ?$ used here.B

Foth sides of the differential equation are multi&lied (% the inte'ratin' factor.

e.'. ) x f ekyedx

dye

kx kx kx =+



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$he L.0.S. of the equation is now equal to the derivative of the &roduct of the inte'ratin'factor and %.

i.e. ) yedx

d yke

dx

dye kx kx kx =+

$he equation is now an e4act differential equation and should (e solved as such.i.e. dx x f e ye

kx kx )∫ =Dinall%* the 'eneral solution is o(tained (% dividin' the equation (% the inte'ratin' factor.

i.e. dx x f ee y kx kx )∫

−=

EXAMPLE

,f x ydx

dy=+ 3 * find y.

0olution

$he inte'ratin' factor is x dx ee 33

=∫ .

Multi&l%in' (% the inte'ratin' factor* we 'et* x x x

xe yedx

dye

3333 =+

,nte'ratin' (oth sides with res&ect to x* we 'et ce xe ye x x x +−= 333

9

1

3

1

+ividin' (% the inte'ratin' factor* we 'et* x ce x y 3

9

1

3

1 −+−= .

EXAMPLE

Solve the differential equation!1 x y

xdx

dy=+ * 'iven that y # 3 when x # !.

0olution

$he inte'ratin' factor is xee xdx

x ==∫ ln1

.

Multi&l%in' (% the inte'ratin' factor* we 'et*3 x y

dx

dy x =+

i.e.3

) x xydx

d =

,nte'ratin' (oth sides with res&ect to x* we 'et c x xy += ::1

>hen x # !* y # 3 !)!:

1)3)!

: =⇒+=∴ cc

!:

1 : +=∴ x xy

+ividin' (% the inte'ratin' factor* we 'et* x

x y !

:

1 3 += .


"


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()*$: the coefficient ofdx

dy must (e 1 in order when the inte'ratin' factor method is

to (e used.

$%ercise !&3&c&c

i)

>orksheet E4act +ifferential Equations ii) Past E4amination =uestions

1. A curve - in the x $ y &lane &asses throu'h the &oint 1*2). At an% &oint x,y) on -*

x e y

dx

dy −=+ .

+a, Dind the 'eneral solution of this differential equation.+b, 0ence find the equation of -* 'ivin' %our answer in the form y = f(x).

[($-.]

!. $he cost Gc of manufacturin' x items ma% (e modelled (% the differentialequation

x cdx

dc12! =+ .

F% usin' a suita(le inte'ratin' factor* solve the differential equation* 'iven thatthere is a cost of G122 when no items are &roduced.


8


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[!""/ Paper 1 – 0 marks]

Se$n& Or&er Differenti! E"#ti$n%

A Equations that ma% (e e4&ressed in the form )!

!

x f dx

yd =

$his t%&e of equation is solved (% inte'ratin'* with res&ect to x* twice.

EXAMPLE

@iven that x dx

yd =

!

!

* find y.

0olution

k cx x y

dx c x dx dx

dy

c x dx

dydx x dx

dx

yd x

dx

yd

++=⇒

+=⇒

+=⇒=⇒=

∫ ∫

∫ ∫

3

!

!

!

!

!

!

A

1

!

1

!

1

where c and k are ar(itrar% constants.

,t should (e noted that the 'eneral solution of a second order differential equation must contain two ar(itrar% constants since it is a second order equation) and

satisfies the equation.

A &articular solution is o(tained (% findin' the values of c and k which satisf% to initialconditions.

$%ercise !&1&c&a

1. Dind the 'eneral solutions of the followin' differential equations

i) "<!

!

−= x dx

yd ii) x

dx

yd !sin

!

!

= iii) x edx

yd =

!

!

!. Solve the differential equation *A3! !!

!

x edx

yd x += 'iven that when x # 2* y # !

and !2=dx

dy.


9


10/17

3. Solve the differential equation 12!

!

=dx

yd * 'iven that when x # H1* y # 3 and when

x # 1* y # 1


11/17

,f the au4iliar% equation has two real equal roots* sa% n1* the 'eneral solution of

the second order differential equation 2!

!

=++ cydx

dyb

dx

yd a is

x ne 1x - y 1) += .

where A and % are ar(itrar% constants.

,f the au4iliar% equation has two com&le4 roots* sa% n1 ± in!* the 'eneral solution

of the second order differential equation 2!

!

=++ cydx

dyb

dx

yd a is

)sincos !!1 x n 1 x n -e y

x n+= *


EXAMPLESolve the followin' second order differential equations:

23!)

29A)

23!)

!

!

!

!

!

!

=++

=+−

=−+

ydx

dy

dx

yd iii

ydx

dy

dx

yd ii

ydx

dy

dx

yd i

0olutions

i) $he au4iliar% equation is n! I !n 3 # 2.

⇒ n I 3)n 1) # 2 ⇒ n # H3 or 1 two real distinct roots)

∴ x x 1e -e y ydx

dy

dx

yd +=⇒=−+

−3

!

!

23!


ii) $he au4iliar% equation is n! n I 9 # 2

⇒ n 3)! # 2 ⇒ n # 3 two real equal roots)

∴ x e 1x - y ydx

dy

dx

yd 3!

!

)29A +=⇒=+−


iii) $he au4iliar% equation is !n! I n I 3 # 2.

⇒ A

!3

A

1

A

!:11 i n ±−=

−±−= two com&le4 roots)


11


12/17

∴

+=⇒=++

− x 1 x -e y y

dx

dy

dx

yd x

A

!3sin

A

!3cos23! A

1

!

!


$%ercise !&1&c&d

D/$0E/ P/E MA$0EMA$,-S EXE/-,SE F

' Equations that ma% (e e4&ressed in the form )!

!

x f cydx

dyb

dx

yd a =++ *

where a* b and c are constants.

$o solve equations of this t%&e we first have to find the 'eneral solution of the equation

2!

!

=++ cydx

dyb

dx

yd a . $his is called the complementar2 function +'4,&

>e now find a particular integral +P, which satisfies the equation

)!

!

x f cydx

dyb

dx

yd a =++ .

A particular integral is an e4&ression which is similar in form to f(x)!

EXAMPLE

i) ,f f (x) is a &ol%nomial of de'ree n* then the P, is a &ol%nomial of de'reen!

ii) ,f f(x) # a sin nx or f(x) = b cos nx or f(x) # a sin nx ± b cos nx* then the P,

is c sin nx I d cos nx (oth tri' functions and I are AL>AJS used).iii) ,f f(x) is an e4&onential function* then the P, is a similar e4&onential

function.


1!


13/17

$he 'eneral solution of the second order differential equation )!

!

x f cydx

dyb

dx

yd a =++ is

the sum of the com&lementar% function and its &articular inte'ral.

i.e. ,f )!

!

x f cydx

dy

bdx

yd

a =++

* then y # -D I P, is the 'eneral solution.

EXAMPLESolve the second order differential equations:

i)

!

+=−+ x ydx

dy

dx

yd

ii) x ydx

dy

dx

yd !sin33!

!

!

=−+

iii) xe ydx

dy

dx

yd "!

!

3:! =−+

0olutions

i) ,n section * e4am&le i)* we found that

x x 1e -e y ydx

dy

dx

yd +=⇒=−+

−3

!

!

23!

∴ -D # x x 1e -e +−3

f(x) # ! x

!

I < is a &ol%nomial of de'ree !.∴ the P, is a &ol%nomial of de'ree !.

∴ let y = ax! I bx & c

⇒ bax dx

dy+= ! and a

dx

yd !

!

!

=

Su(stitutin' for y and its derivatives in the 'iven equation* we 'et that

Equatin' the coefficients of like term* we 'et that

3

!!3 −=⇒=− aa

9

8

3

:

23:

−=∴

=⇒=−

b

a

bba

!"

"3

3

−=∴

−+=⇒=−+

c

baccba


13


14/17

∴ P, #!"

"3

9

8

3

! ! −−− x x

and the 'eneral solution is

!"

"3

9

8

3

! !3 −−−+= − x x 1e -e y x x .

ii) x ydx

dy

dx

yd !sin33!

!

!

=−+

Drom a(ove* -D # x x 1e -e +−3

Since f(x) # 3 sin ! x* let y # x b x a !cos!sin +

⇒ x b x adx

dy!sin!!cos! −= and x b x a

dx

yd !cos:!sin:

!

!

−−=

Su(stitutin' in the 'iven equation* we 'et that

x b x a !cos:!sin: −− I ! x b x a !sin!!cos! − ) 3 x b x a !cos!sin + ) ≡ 3 sin ! x

Equatin' coefficients of like terms* we 'et thatH"a b # 3 and a "b # 2.Solvin' these equations simultaneousl%* we 'et that

A:

:!"−=a and

1A

A1−=b

∴ P, # x x !cos1A

A1!sin

A:

:!"−−

∴ $he required 'eneral solution is

= y x x 1e -e +−3 x x !cos1A

A1!sin

A:

:!"−−

iii) xe ydxdy

dx yd "!

!

3:! =−+

Drom a(ove* -D # x x 1e -e +−3

Since xe x f "3) = * let xCe y "=

xCedx

dy ""=⇒ and xCedx

yd "!

!

:9=

( (


15/17

$%ercise !&1&c&1!c

D/$0E/ P/E MA$0EMA$,-S EXE/-,SE -

1. @iven that A, %, ' and c are constants* verif% that the 'eneral solution of:

a) !sincos k

c 2x 1 2x - y ++= satisfies the differential equation

.!

!

!

c y 2

dx

yd =+

()!k

c 1e -e y 2x 2x −+= − satisfies the differential equation .!

!

!

c y 2 dx

yd =−

C$his t%&e of differential equation occurs in a num(er of scientific situations.B

1


16/17

a) !:

1!

!

=+ ydx

yd 'iven that when 1*

:== y x

and when 1*!

−== y x

.

() !!

!

=− ydx

yd 'iven that when !*2 == y x and A=

dx

dy.

c) ::

1!

!

=+ ydx yd

'iven that when 19* == y x and 1−=dx

dy

.

d) ::!

!

=− ydx

yd 'iven that when !*2 ==

dx

dy x and

!

3!

!

=dx

yd .

+E/S$A+,@ P/E MA$0EMA$,-S EXE/-,SE !2E =ES$,?S $? 9

D/$0E/ P/E MA$0EMA$,-S EXE/-,SE -

S)5*)( )4 744$8$(*-5 $-*)(S . S.S***)(

i-!i$'r./

Durther Pure Mathematics (% Frian and Mark @aulter

nderstandin' Pure Mathematics (% A.K. Sadler and +.>.S. $hornin'


1


17/17

-alculus with Anal%tic @eometr%

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