4) Int´ egrer les syst` emes diff´ erentiels suivants :

Utilisation de la transform´ ee de Laplace

1) D´ eterminer les transform´ ee de Laplace suivantes :

a) L [sin(3t)e^−tU(t)] b) L [(2t+ 1)e^−3tU(t)]

2) D´ eterminer les originaux suivants :

a) L⁻¹

p p²+ 4p+ 6

b) L⁻¹ p

(p+ 1)²

c) L⁻¹

p p²+ 2p+ 5

3) Int´ egrer les ´ equations diff´ erentielles suivantes :

a) x⁰⁰(t) + 3x⁰(t) + 2x(t) = 0 avec x(0) = 1 ; x⁰(0) = 0 b) x⁰⁰(t) + 6x⁰(t) + 9x(t) = e^−2t U(t) avec x(0) = 0 ; x⁰(0) = 0 c) x⁰⁰(t)−x(t) = (3e^2t+t²+ 1) U(t) avec x(0) = 0 ; x⁰(0) = 1 d) x⁰⁰(t)−4x(t) = (3e^−t−t²) U(t) avec x(0) = 0 ; x⁰(0) = 1

4) Int´ egrer les syst` emes diff´ erentiels suivants :

a)

x⁰(t) =x(t) + 5y(t)

y⁰(t) = x(t)−3y(t) avec

x(0) = 1 y(0) = 2 b)

x⁰(t) =−7x(t)−6y(t) +tU(t)

y⁰(t) = 12x(t) + 10y(t) avec

x(0) = 0 y(0) = 0 c)







x⁰(t) = 5x(t) +y(t)−z(t) y⁰(t) = 2x(t) + 4y(t)−2z(t) z⁰(t) = x(t)−y(t) + 3z(t)

avec







x(0) = 1 y(0) = 0 z(0) = 0 d)

x⁰(t) =−7x(t)−6y(t)

y⁰(t) = 12x(t) + 10y(t) avec

x(0) = 0 y(0) = 1 e)







x⁰(t) = x(t) +y(t)−z(t) y⁰(t) =x(t)−y(t) +z(t) z⁰(t) = −x(t) +y(t) +z(t)

avec







x(0) = 1 y(0) = 0 z(0) = 3

Transform´ ee de Laplace (Solutions)

1) D´ eterminer les transform´ ee de Laplace suivantes :

a) L [sin(3t)e^−tU(t)]

L [sin(3t)U(t)] = 3 p²+ 3² L

sin(3t)e^−tU(t)

= 3

(p+ 1)²+ 3² = 3 p²+ 2p+ 10 b) L [(2t+ 1)e^−3tU(t)]

L [(2t+ 1) U(t)] = 2 p² +1

p = p+ 2 p² L

(2t+ 1)e^−3tU(t)

= (p+ 3) + 2

(p+ 3)² = p+ 5 p²+ 6p+ 9

2) D´ eterminer les originaux suivants :

a) L⁻¹

p p²+ 4p+ 6

p

p²+ 4p+ 6 = (p+ 2)−2

(p+ 2)²+ 2 = p+ 2 (p+ 2)²+ (√

2)² −

√2√ 2 (p+ 2)²+ (√

2)² L⁻¹

"

p p²+ (√

2)² +

√2 p²+ (√

2)²

#

= cos(√

2t) + sin(√ 2t)

U(t) L⁻¹

p p²+ 4p+ 6

=

cos(√

2t)−√

2 sin(√ 2t)

e^−2tU(t)

b) L⁻¹ p

(p+ 1)²

p

(p+ 1)² = (p+ 1)−1

(p+ 1)² = 1

p+ 1 − 1 (p+ 1)² L⁻¹

1 p − 1

p²

= (1−t)U(t) L⁻¹

p (p+ 1)²

= 1−t

e^−tU(t)

c) L⁻¹

p p²+ 2p+ 5

p

p²+ 2p+ 5 = (p+ 1)−1

(p+ 1)²+ 2² = p+ 1

(p+ 1)²+ 2² −1 2

2 (p+ 1)²+ 2² L⁻¹

p p²+ 2p+ 5

=

cos(2t)− 1

2sin(2t)

e^−t U(t)

3) Int´ egrer les ´ equations diff´ erentielles suivantes :

a) x⁰⁰(t) + 3x⁰(t) + 2x(t) = 0 avec x(0) = 1 ; x⁰(0) = 0 (p²X(p)−p−0) + 3(pX(p)−1) + 2X(p) = 0

(p² + 3p+ 2)X(p) =p+ 3 X(p) = p+ 3

p²+ 3p+ 2 = p+ 3 (p+ 1)(p+ 2) X(p) = 2

p+ 1 − 1 p+ 2 x(t) = 2e^−t−e^−2t

U(t)

b) x⁰⁰(t) + 6x⁰(t) + 9x(t) = e^−2t U(t) avec x(0) = 0 ; x⁰(0) = 0 (p²X(p)−0−0) + 6(pX(p)−0) + 9X(p) = 1

p+ 2 (p²+ 6p+ 9)X(p) = 1

p+ 2

X(p) = 1

(p+ 2)(p+ 3)² X(p) = 1

p+ 2 + 1

(p+ 3)² − 1 p+ 3 x(t) = e^−2t−(t+ 1)e^−3t

U(t)

c) x⁰⁰(t)−x(t) = (3e^2t+t²+ 1) U(t) avec x(0) = 0 ; x⁰(0) = 1 (p²X(p)−0−1)−X(p) = 3

p−2 + 2 p³ +1

p (p²−1)X(p) = 3

p−2 + 2 p³ +1

p + 1 X(p) = p⁴+ 2p³−2p²+ 2p−4

p³(p−2)(p²−1) X(p) = 1

p−2 +

1 2

p−1 +

3 2

p+ 1 − 2 p³ − 3

p x(t) =

e^2t+1 2e^t+ 3

2e^−t−t²−3

U(t)

(p²X(p)−0−1)−4X(p) = 3

p+ 1 − 2 p³ (p²−4)X(p) = 3

p+ 1 − 2 p³ + 1 X(p) = p⁴+ 4p³−2p−2

p³(p+ 1)(p²−4) X(p) =

7 16

p−2 +

7 16

p+ 2 − 1 p+ 1 +

1 2

p³ +

1 8

p x(t) =

7

16e^2t+ 7

16e^−2t−e^−t+ 1 4t²+1

8

U(t)

4) Int´ egrer les syst` emes diff´ erentiels suivants :

a)







x⁰(t) = x(t) + 5y(t) y⁰(t) =x(t)−3y(t)

avec







x(0) = 1 y(0) = 2







pX(p)−1 = X(p) + 5Y(p) pY(p)−2 = X(p)−3Y(p)







(p−1)X(p) − 5Y(p) = 1

−X(p) + (p+ 3)Y(p) = 2















X(p) = p+ 13 p² + 2p−8 =

5 2

p−2 −

3 2

p+ 4 Y(p) = 2p−1

p² + 2p−8 =

1 2

p−2 +

3 2

p+ 4















x(t) = 5

2 e^2t−3 2 e^−4t

U(t)

y(t) = 1

2e^2t+3 2 e^−4t

U(t)

b)

x⁰(t) =−7x(t)−6y(t) +tU(t)

y⁰(t) = 12x(t) + 10y(t) avec

x(0) = 0 y(0) = 0







pX(p) = −7X(p)−6Y(p) + 1 p² pY(p) = 12X(p) + 10Y(p)







(p+ 7)X(p) + 6Y(p) = 1 p²

−12X(p) + (p−10)Y(p) = 0











X(p) = p−10

p⁴−3p³+ 2p² = 9

p−1 − 2

p−2 − 5 p² − 7

p

Y(p) = 12

p⁴−3p³+ 2p² = − 12

p−1 + 3

p−2 + 6 p² +9

p







x(t) = 9e^t−2e^2t−5t−7 U(t) y(t) = −12e^t+ 3e^2t+ 6t+ 9

U(t)

c)



y⁰(t) = 2x(t) + 4y(t)−2z(t) z⁰(t) = x(t)−y(t) + 3z(t)

avec



y(0) = 0 z(0) = 0







pX(p)−1 = 5X(p) +Y(p)−Z(p) pY(p) = 2X(p) + 4Y(p)−2Z(p) pZ(p) = X(p)−Y(p) + 3Z(p)







(p−5)X(p) − Y(p) + Z(p) = 1

−2X(p) + (p−4)Y(p) + 2Z(p) = 0

−X(p) + Y(p) + (p−3)Z(p) = 0



















X(p) = p−5

(p−6)(p−4) =

1 2

p−4 +

1 2

p−6

Y(p) = 2

(p−6)(p−2) =

1 2

p−6 −

1 2

p−2

Z(p) = 1

(p−2)(p−4) =

1 2

p−4 −

1 2

p−2























x(t) = e^4t+e^6t 2 U(t) y(t) = e^6t−e^2t

2 U(t) z(t) = e^4t−e^2t

2 U(t)

d)

x⁰(t) =−7x(t)−6y(t)

y⁰(t) = 12x(t) + 10y(t) avec

x(0) = 0 y(0) = 1 pX(p) = −7X(p)−6Y(p)

pY(p)−1 = 12X(p) + 10Y(p)

(p+ 7)X(p) + 6Y(p) = 0

−12X(p) + (p−10)Y(p) = 1











X(p) = −6

p³−3p²+ 2p = 6

p−1 − 6 p−2 Y(p) = p+ 7

p³−3p²+ 2p = 9

p−2 + 8 p−1







x(t) = 6 e^t−e^2t U(t) y(t) = 9e^2t−8e^t

U(t)

e)







x⁰(t) = x(t) +y(t)−z(t) y⁰(t) =x(t)−y(t) +z(t) z⁰(t) = −x(t) +y(t) +z(t)

avec







x(0) = 1 y(0) = 0 z(0) = 3







pX(p)−1 = X(p) +Y(p)−Z(p) pY(p) = X(p)−Y(p) +Z(p) pZ(p)−3 = −X(p) +Y(p) +Z(p)







(p−1)X(p) − Y(p) + Z(p) = 1

−X(p) + (p+ 1)Y(p) − Z(p) = 0 X(p) − Y(p) + (p−1)Z(p) = 3



















X(p) = p²−3p−2 p³−p²−4p+ 4 =

4 3

p−1− 1 p−2+

2 3

p+ 2

Y(p) = 4

(p−1)(p+ 2) =

4 3

p−1−

4 3

p+ 2 Z(p) = 3p²−p−6

p³−p²−4p+ 4 =

4 3

p−1+ 1 p−2+

2 3

p+ 2























x(t) = 1 3

4e^t−3e^2t+ 2e^−2t

U(t)

y(t) = 4 3

e^t−e^−2t U(t)

z(t) = 1 3

4e^t+ 3e^2t+ 2e^−2t U(t)