Akademin för Informationsteknologi - ITE MA2047 Algebra och diskret matematik 1 / 1

Ex 6 Los

X y 't 2y = I

{ya , -

- z

×

Xy't Ly =

' ⇒ y

'e Ey =

x 2

Mutt . ehv in . den integrate fuhtrvh EG

e

214×1 21 #

= elnx?× ?

y'

. x 't y - 2x = In x ⇒ y - x

'

= flux d x = f I . In xd x = xlnx - J x . # dx

- f g F g Fg

'

= D C y . X Y= X In x - X t C

Allin .Is g : y Cx ) =

I II - tf t Iz

y lil =

' ÷ - I t f- = C - I = 2 ⇒ C =3

.

'

.

Sont Inning : y Cx ) =

'

II - I e- ¥ .

Akademin för Informationsteknologi - ITE MA2047 Algebra och diskret matematik 1 / 1

Ex 7 Los7

x y'

- Ly t,÷=o ,x a o

{yen -

- o

3

x y'

- rye ,I×÷=o ⇐ y'

- Ey =- ¥ ,

Mutt.

eh . in. int . frht . et " '

= e- 21 ht "

= e- " " ×

= em ×- I ¥ ,

:T

X ? O

y'

. # t y - I - ⇒ =D ( y . I . ) =- ¥ ,

⇐ y . In = f - ¥ ,dx = - If

'

dx =- flu lltxltc = - I In Cltx ) -1C

f-

⇐ ycx ) = XI( C - In lit x' I ) ← All m .

I say .

Ycl ) = I ( C - In 2) = O ⇐ C = In 2

-

-

'

.

Srt Isg : y L x ) = ( In 2 - lull -1×4 ) .

Akademin för Informationsteknologi - ITE MA2047 Algebra och diskret matematik 1 / 1

Ex 10 Los

y'

= y -4it x e- ×

){ yeol = I

y'

= y 'Ll txe- ×

I faff,

= I exe

- ×

⇐ ftyady - flexiTtx-

gcy ) y' hlx )

⇐ - f- = S4exe

-

Mdx = flax + Sxixdx = x + x. C - e- "

) - GIC - e -5 dx

g f g F g' F

= x - xe-Xt f e- ×

dx = x - xe- ×

- e-

it C = - e

-Flex ) txt C

⇒ ycx ) = # ← Allin.

Isge- ' '

( itx ) - x - C

yeol =1- =

1- = I ⇐ C = o

e°( Ito ) - o - cI - C

.

'

.

sort Isg : y Cx I =

1-e-

4 It x ) - x

-

Akademin för Informationsteknologi - ITE MA2047 Algebra och diskret matematik 1 / 1

Ex 11 Lots y' tglxly =h Cx )

( Lt Xyy ' = x' y

'

,× > o

→ gly ) y'

= h ( X )

{finny " ' ' '

a

lit xyy'

-- x' y

'

⇐ ÷ If,

= It ⇐ Sfpd , =/ ,Y÷dxgly ) y

' hlx )

⇐ - I -

- I ,÷.dx -

- S dx - KiI- ¥)dx

= 14- ¥ )dx = x - avcknx t C

⇐ y Cx ) = 1- ← Allan . I Sg

arclznx - x - C

ycx ) =1- - → - I = I de x - Ot ⇐ C = - I

avctanx - X - C-

a→ 0 6

.

'

.

Sunt. lig : ycx7=an,+T -

Akademin för Informationsteknologi - ITE MA2001 Envariabelanalys 1 / 1

Ex Los → y'

tgc # ly = hlx )

{ If,

'

,

+ ¥-

- o.

" °

gamy 's next

xy'

- ye ,÷µ= o ⇐ Y'

- TY = - ×¥y-

-g Cx )

u Lx )

Mutt. ehv med den ihteguwnle felton e

G' "= e-

lhlx ! -In ×

in x- '

e = C= x

-

If :

T

y'

. It y . ( - fi ) = -

1-no

xnclexy-

=DI y . I )⇐ i . t.es#*ix-- - I x -

-- S -

×¥÷⇒dx --

= 1¥,

- IT)d× = arctan x tf t C

⇐ y Cx ) = x anchor x t I t Cx ← Allin.

Isg .

y Cl ) = I - a - c tant t I t C = YI t I t C = I ⇐ C = - YI

-

'

.Smt Isg : y

C x ) = xarctunx + I -

.

Akademin för Informationsteknologi - ITE MA2001 Envariabelanalys 1 / 1

Ex Losy

'+ gcxly -

-

hee )x - y

y'

-_ x e

→ g ly ) yl = hlx ){

yeol -

-o

e Ex

y'

= x ex- Y

= x ex . e

- Y EY

.

y'

= × ex ⇐ feYdy=fxe×dx- -

g ly ) hlx )

⇐ f e't

dy = EY = fxexdx = x ex - fi . exdx = x ex- exec

g f g F g'

F

⇒ ylx ) = In ( x ex - e 't C ) ← Allin.

lsg .

yeol =In ( o . e° - e°t C ) = In ( C - I ) = o ⇐ C - I = I ⇐ C = 2

,

.

-

.Smt by : ycx ) = In ( e' ' Cx- 1) +2 ) .

Akademin för Informationsteknologi - ITE MA2001 Envariabelanalys 1 / 1

glx )EX Los

y'

-27 y = 2×3 → y'

c- gcxly = hlx )

{yco ) = o { gly ) y

' = hlx )

Mutt ehr m . den inkgunnde foh torn e

G ↳ I= e-

× ?

y!

e- × I ye

- × ? C-2x ) = 2×3 e- ×

'

-D ly . e

- x )⇒ y . e-

i= f 2x

'

e

- × 'd x = - fxg? eight,⇒ x' . e- If 2×1 - E) dx

= - x' e

' × I fzxe- "

dx =- x' e

- I e-

× I c

⇐ y C x ) = - x

'

- I t C ex"

← Allin Isg

y co ) = o - I t C e

°

= C - I = o ⇐ c = I

-

'

.

sort Csg : ycxl = e

"- x

'

- I.

Akademin för Informationsteknologi - ITE MA2001 Envariabelanalys 1 / 1

Ex Lss y'

= 2x ( I - y ) → y'

tgCx7y=hLx ) ( 11

{yco , = 2

" i " !→ { gly ) y

' =hCx ) H

( I ) y'

= 2x ( I - y ) ⇐ y 't 2x y = 2x-

glx )

Mutt. ehv

. in. int

.hut

. EACH= ex

"

:

y'

. ex 't yet ? 2x = 2x ex

"

-=D C y . ex

'

)⇒ y - ex

'

= f 2x ex 'dx = ex I C

- x'

⇐ ylx ) = I + Ce ← Allin. Isg

yco ) = It Ceo = It C = 2 ⇒ C= f

.

'

.

Sont Isg : ycxl = Ite' × ?

Akademin för Informationsteknologi - ITE MA2001 Envariabelanalys 1 / 1

(2) y'

= excl - y ) ⇐'

=

" '

* ops ! y-

- I a- nee Isg !

T T Det at inte den vi so . her

Oury t I dy- efte - son y co ) = 2 !

DX

⇐ f ÷y dy = f 2x dx

⇐ - Intl - yl = x 't C ⇐ In I I - yl- '

= x 't C

⇐ Lull - Yi ! ex't C

= ex ? ec = De

"

-

=D > O

⇐ ¥ ,= Dex

"

⇐ ÷ = Ibex'

= E. ex'

E to

⇐ I - y = ÷× ,= ¥ - e

- ×"

⇐ ycx ) =I - F. e

' ×"

← Allin log !

( I F=o qer 4=1 ovan )

'

y Lo ) = I - Feo = 2 ⇐ F = - I ⇐

.

- x2

- .Solt 15g : ylx ) = It e