Problem Set 6.7 No. 1

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Advanced Engineering Mathematics, by Erwin Kreyszig 10^th. Ed.

Problem Set 6.7

No. 1

(2)

(3)

(4)

(5)

No. 2

y₁^'−y₂=0, y₁+y₂^'=2cos t , y₁(0)=1, y₂(0)=0

Writing ℒ {^y1}^=Y1(s),

ℒ {^y2}^=Y2(s) sY₁(s)−y₁(0)−Y₂(s)=1

Y₁(s)+sY₂(s)−y₂(0)= ^{2 s}

s²+1 Replace _y₁(0)=1, y₂(0)=0 sY₁(s)−Y₂(s)=1

Y₁(s)+sY₂(s)= ^{2 s}

s²+1

× s＋ (s²+1)Y₁( s)=s+ ^2s

s²+1

Y₁(s )= ^s

s²+1+ 2 s

(_s²₊₁)²

Then from Y₂(s)=sY₁(s )−1= ^s²

s²+1+ 2s²

(s²+1)²⁻¹⁼⁻

1

s²+1+ 2s²

(s²+1)²

Using (3) of sec.6.6 ℒ^-1

{

⁽^s²^+β^s ²⁾²

}

⁼2β^t sin βt

y₁(t )=cost +^{2 t}₂ sin t=cost+t sin t In this case β=1

And using (4) of sec.6.6 ℒ^-1

{

⁽^s²⁺^s²^β²⁾²

}

⁼^{2 β}¹ ⁽sin βt+βt cosβt)

y₂(t )=−sin t+²₂(sin t+t cost )=−sin t+sin t+t cost=t cost In this case β=1

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No. 3

y₁^'−2 y₁+3 y₂=0, y₂^'−y₁+2 y₂=0, y₁(0)=1, y₂(0)=0

ℒ {^y2}^=Y2(s) sY₁(s)−y₁(0)−2 Y₁(s)+3 Y₂(s)=0 sY₂(s)−y₂(0)−Y₁(s)+2 Y₂(s)=0

Replace ^y¹(0)=1, y₂(0)=0 (s−2)Y₁(s)+3 Y₂(s)=1

−Y₁(s)+(s+2)Y₂(s)=0

×( s+2)－ × 3 (s²−4 +3)Y₁( s)=s+ 2

Y₁(s)= ^s+2

s²−1 Then from Y₂(s)=_s+2¹ Y₁(s)= ¹

s²−1

Furthermore Y₁(s)=^s+2

s²−1=

3 2

s−1−

1 2

s+1

Y₂(s)= ¹

s²−1=

1 2

s−1−

1 2

s+1

y₁(t )=³₂e^t−¹₂e^−t=¹₂e^t+¹₂e^−t+2(¹2e^t−¹₂e^−t)=cosht +2 sinh t

y₂(t)=¹₂e^t−¹₂e^−t=sinht

No. 4

y₁^'=4 y₂−8cos 4t , y₂^'=−3 y₁−9sin 4t , y₁(0)=0, y₂(0)=3

ℒ {^y2}^=Y2(s)

sY₁(s)−y₁(0)=4Y₂(s)− ^{8 s}

s²+16

sY₂(s)−y₂(0)=−3Y₁(s)− ³⁶

s²+16 Replace _y₁(0)=0, y₂(0)=3

sY₁(s)−4Y₂(s)=− ^{8 s}

s²+16

3Y₁(s)+sY₂(s)=3− ³⁶

s²+16

(7)

× s＋ × 4 (^s²⁺¹²)^Y₁( s)=− ^{8 s}²

s²+16+12−144 s²+16

(s²+12)Y₁( s)=^−8s²^{+12 s}²^+192−144

s²+16 (s²+12)Y₁(_s)₌^{4 s}²⁺⁴⁸

s²+16=4(s²+12)

s²+16

Y₁(s)= ⁴

s²+16

×3－ × s

(−12−s²)Y₂(_s)₌₋ ^{24 s}

s²+16−3 s+36 s

s²+16=−24 s−3 s³−48s+36s

s²+16 =−36 s−3s³

s²+16 =3s(−12−s²)

s²+16 Y₂(s)= ^{3 s}

s²+16 y₁(t)=sin 4 t y₂(t)=3 cos4 t No. 5

y₁^'=y₂+2−u(t−1) , y₂^'=−y₁+1−u(t−1), y₁(0)=1, y₂(0)=0

ℒ {^y2}^=Y2(s)

sY₁(s )− y₁(0 )=Y₂( s)+²_s−^e^−s s sY₂(s )− y₂(0 )=−Y₁( s)+¹_s−^e^−s

s ^Replace ^y1(0)=1, y₂(0)=0

sY₁(s )−Y₂(s )=1+²_s−^e^−s s Y₁(s )+ sY₂(s )=¹_s−^e^−s

s

× s＋ (s²+1)Y₁(s)=s +2−e^−s+¹_s−^e^−s

s =2+s²+1

s −(^s+1s )^e^−s

Y₁(s)= ²

s²+1+1

s− s+1 s(s²+1)^e

−s

= ²

s²+1+1

s−

(

¹^s⁻s^s−1²+1

)

^e^−s⁼s²²+1+1

s−

(

¹^s⁻s²^s+1+ 1 s²+1

)

^e^−s

－ × s

(8)

(−1−s²)Y₂(s)=1+²_s−^e^−s

s −1+e^−s=2 s−e^−s

s +e^−s

Y₂(s)=− ²

s(_s²₊₁)+

[

⁻^s²¹⁺¹⁺^s⁽^s¹²⁺¹⁾

]

^e^−s⁼⁻²^s ⁺^s^2s²⁺¹⁺

⁽

⁻^s²¹⁺¹⁺¹^s⁻^s²^s⁺¹

⁾

^e^−s

y₁(t)=1+2 sint−[^1−cos⁽^t−1⁾^+sin⁽^t−1⁾]^u⁽^t−1⁾

y₂(t)=−2+2 cos t+[^1−sin⁽^{t −1}⁾^−cos⁽^t−1⁾]^u⁽^t−1⁾

No. 6

y₁^'=5 y₁+y₂, y₂^'=y₁+5 y₂, y₁(0)=1, y₂(0)=−3

ℒ {^y2}^=Y2(s) sY₁(s)−y₁(0)=5 Y₁(s)+Y₂(s)

sY₂(s)−y₂(0)=Y₁(s)+5 Y₂(s)

Replace ^y¹(0)=1, y₂(0)=−3 sY₁(s)−5 Y₁(s)−Y₂(s)=1 (^s−5)Y₁(s)−Y₂(s)=1

−Y₁(s)+sY₂(s)−5 Y₂(s)=−3 ^−Y1(s)+(s−5)Y₂(s)=−3

×( s－5)＋ [⁽^s−5⁾²⁻¹]^Y₁⁽^s⁾^=s−5−3

Y₁(s)= ^s−8

(s²−10s+24)= 2

s−4− 1 s−6

+ ×( s－5)

[⁻¹⁺⁽^s−5⁾²]^Y₂⁽^s⁾=1−3s+15=−3 s+16

Y₂(s)= ^{−3 s+16}

(s²−10 s+24)= −2 s−4− 1

s−6 y₁(t )=2e^4t−e^6t

y₂(t )=−2e^{4 t}−e^6t

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No. 7

y₁^'=2 y₁−4 y₂+u(t−1) e^t, y₂^'=y₁−3 y₂−u(t−1) e^t, y₁(0)=3, y₂(0)=0

ℒ {^y2}^=Y2(s)

sY₁(s )− y₁(0 )=2 Y₁(s )−4 Y₂( s)+e ^e^−s s−1 sY₂(s )− y₂(0 )=Y₁(s )−3 Y₂( s )−e ^e^−s

s−1 ^Replace ^y1(0)=3, y₂(0)=0

sY₁(s )−2 Y₁( s)+4 Y₂(s )=3+e ^e^−s

s−1 (s−2) Y₁( s)+4 Y₂(s)=3+ e ^e^−s s−1

−Y₁(s )+sY₂(s )+3 Y₂(s )=−e ^e^−s

s−1 −Y₁(s )+( s+3) Y₂( s)=−e ^e^−s s−1

× (s＋3)－×4 [(s−2) ( s+3)+ 4]^Y₁^{( s)=}

(

^3+es−1^e^−s

)

( s+3 )+ 4 e e^−s s−1

(s²+s−2)Y₁(s )=3 s+9+(s−1^s+7)^ee^−s

Y₁(s )= ^{3 s+9}

(s−1) (s+2)+ ⁽^s+7⁾

(s−1)²(s+2)ee^−s= 4

s−1− 1

s+2+

[

⁻^(s−1)⁵⁹^s+²²⁹⁹ ⁺^s+2⁵⁹

]

^ee^−s

=_s−1⁴ −_s+2¹ +

[

^(s−1)⁻⁵⁹ ⁺^(s−1)⁸³ ²⁺^s+2⁵⁹

]

^ee^−s

＋×( s－2) [^{4 +}⁽^{s +3}^{) (}^s−2⁾]^Y₂⁽^s⁾^=3+e_s−1ê^−s ⁻⁽^s−2⁾ê ê

−s

s−1

(s²+s−2)Y₂(s)=3−(s−3)e ^e^−s s−1

Y₂(s)=₍_s−1³_{) (}_s+2₎− ^s−3

(s−1)²(s+2)ee^−s= 1

s−1− 1

s+2−

[

^{( s−1)}⁵⁹^s−¹¹⁹²⁻^s+2⁵⁹

]

^ee^−s

=_s−1¹ −_s+2¹ −

[

^s−1⁵⁹ ⁻^{( s−1)}²³ ²⁻^s+2⁵⁹

]

^ee⁻^s

(10)

y₁(t )=4 e^t−e^−2t+[⁻⁵9ee^(t−1)+⁸₃(t−1)ee^{( t−1)}+⁵₉ee⁻^2(t−1)]^u(t−1)

=4 e^t−e^−2t+[⁽⁸³^t−²⁹⁹ ⁾^e^t⁺⁵⁹^e⁽^{−2t +3}⁾]^{u (t−1)}

y₂(t)=e^t−e^−2t−[⁵⁹êe^t−1⁻²³⁽^t−1⁾êe^t−1⁻⁵⁹êe^{−2 (t−1)}]û⁽^t−1⁾

=e^t−e^{−2 t}−[⁽⁻²³^t+¹¹⁹ ⁾^e^t⁻⁵⁹^e⁽^{−2 t+3}⁾]^{u (t−1)}

=e^t−e^{−2 t}+[⁽²³^t−¹¹⁹ ⁾^e^t⁺⁵⁹^e⁽^−2t+3⁾]^{u(t−1 )}

No. 8

y₁^'=−2 y₁+3 y₂, y₂^'=4 y₁−y₂, y₁(0)=4 , y₂(0)=3

ℒ {^y2}^=Y2(s) sY₁(s)−y₁(0)=−2 Y₁(s)+3 Y₂(s) sY₂(s)−y₂(0)=4 Y₁(s)−Y₂(s)

Replace ^y¹(0)=4, y₂(0)=3 sY₁(s)+2 Y₁(s)−3 Y₂(s)=4 (^{s +2})Y₁(s)−3 Y₂(s)=4

−4 Y₁(s)+sY₂(s)+Y₂(s)=3 ^{−4 Y}1(s)+(s +1)Y₂(s)=3

×( s＋1)＋×3 [⁽^{s +2}^{) (}^{s+ 1}⁾⁻¹²]^Y₁⁽^s⁾^{=4 s +4 +9}

Y₁(s)= ^{4 s+13}

(s²+3 s−10)= 3 s−2+ 1

s+5

×4+×( s＋2)

[⁻¹²⁺⁽^{s +2}^{) (}^{s+ 1}⁾]^Y₂⁽^s⁾=16+3 s+6=3 s +22

Y₂(s)= ^{3 s+22}

(s²+3 s−10)= 4

s−2− 1 s+5 y₁(t )=3e^2t+e^−5t

y₂(t )=4 e^{2 t}−e^{−5 t}

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No. 9

y₁^'=y₁+y₂, y₂^'=−y₁+3 y₂, y₁(0)=1, y₂(0)=0

ℒ {^y2}^=Y2(s) sY₁(s)−y₁(0)=Y₁(s)+Y₂(s)

sY₂(s)−y₂(0)=−Y₁(s)+3 Y₂(s)

Replace ^y¹(0)=1, y₂(0)=0 sY₁(s)−Y₁(s)−Y₂(s)=1 (^s−1)Y₁(s)−Y₂(s)=1 Y₁(s)+sY₂(s)−3 Y₂(s)=0 ^Y1(s)+(s−3)Y₂(s)=0

×( s－3)＋ [⁽^s−1^{) (}^s−3⁾⁺¹]^Y₁⁽^s⁾^=s−3

Y₁(s)= ^s−3

(s²−4 s+4)=(s−2)−1 (s−2)² ⁼

1

s−2− 1 (s−2)²

－×( s－1)

[⁻¹⁻⁽^s−3^{) (}^s−1⁾]^Y₂⁽^s⁾⁼¹ Y₂(s )=− ¹

(s²−4 s+4)=− 1 (s−2)²= y₁(t )=e^2t−te^{2 t}

y₂(t )=−te^2t

No. 10

y₁^'=−y₂, y₂^'=−y₁+2[^{1−u(t−2π )}]^{cost , y}₁^{(0)=1, y}₂⁽⁰⁾⁼⁰

ℒ {^y2}^=Y2(s)

2[1−u (t−2 π )]cost=2 cost−2 u (t−2 π ) cos(t−2 π ) sY₁(s)−y₁(0)=−Y₂(s)

(12)

sY₂(s )− y₂(0 )=−Y₁( s)+ ^{2 s}

s²+1−2 se^{−2 πs}

s²+1 Replace ^y1(0)=1, y₂(0)=0

sY₁(s)+Y₂(s)=1 Y₁(s )+ sY₂(s )= ^{2 s}

s²+1−2 se^{−2 πs} s²+1

×s－ (s²−1)Y₁(s )=s− ^{2 s}

s²+1+2 se^{−2 πs} s²+1

Y₁(s)= ^s

(_s²₋₁)− 2 s

(^s²⁻¹)(^s²⁺¹)⁺

2se^{−2 πs}

(^s²⁻¹) (^s²⁺¹)

=

1 2

s−1+

1 2

s+1− s

s²−1+ s

s²+1+

(

s²−1^s − s

s²+1

)

^e^−2πs

=

1 2

s−1+

1 2

s+1−

1 2

s−1−

1 2

s+1+ s

s²+1+

(

^s−1¹² ⁺^s+1¹² ⁻s²^s+1

)

^e^−2πs

= ^s

s²+1+

(

^s−1¹² ⁺^s+1¹² ⁻s²^s+1

)

^e^{−2 πs}

－×s

(1−s²)Y₂(_s)₌₁₋ ^2s²

s²+1+2 s²e^{−2 πs}

s²+1 =−(s²−1)

s²+1 +2s²e^−2πs s²+1

Y₂(s)= ¹

s²+1− 2s²e^{−2 πs}

(^s²⁻¹)(^s²+1)

= ¹

s²+1−

(

s²−1¹ + 1

s²+1

)

^e^{−2 πs}

= ¹

s²+1−

(

^s−1¹² ⁻^s+1¹² ⁺s²¹+1

)

^e^−2πs

y₁(t )=cos t +[¹²^e^{( t−2 π)}⁺¹²^e^{−(t −2 π )}−cos(t−2 π )]^{u (t−2 π )}

=cos t+[cosh (t−2 π )−cos t]u (t−2 π )

(13)

y₂(t )=¹₄e^t−¹₄e^−t+¹₂sint−[¹²^e^{( t −2 π )}⁻¹²^e^{−(t−2 π )}+sin (t−2 π )]^{u (t−2 π )}

=sin t−[sinh (t−2 π )+sin t]^{u (t−2 π )}

No. 11

y₁} } =y rSub { size 8{1} } +3y rSub { size 8{2} } ,y rSub { size 8{2} } rSup { size 8{

=4 y₁−4e^t, y₁(0)=2, y₁^'(0)=3, y₂(0)=1, y₂^'(0)=2

ℒ {^y2}^=Y2(s)

s²Y₁(s)−sy₁(0)− y₁^'(0)=Y₁(s)+3Y₂(s)

s²Y₂(_s)_−sy₂(₀)−y₂^'(₀)_=4Y₁(_s)−_s−1⁴ _Replace

y₁(0)=2, y₁^'(0)=3, y₂(0)=1, y₂^'(0)=2 (s²−1)Y₁(s)−3 Y₂( s)=2 s+3

−4Y₁(_s)+s²Y₂(_s)_=s+2−_s−1⁴

× s² _＋×3

[^s²⁽^s²⁻¹⁾⁻¹²]^Y₁⁽^s⁾⁼⁽^2s+3⁾^s²^+3s+6−_s−1¹² ^{=2 s}³⁺^3s²^+3s+6−_s−1¹²

Y₁(s)=^{2 s}³^{+3 s}²^{+3 s+6} s⁴−s²−12−12

(s⁴−s²−12)(s−1)

= ^{2 s}³^{+3 s}²^{+3 s+6}

(s²−4)(s²+3)⁻

12

(s²−4)(s²+3)(s−1)

=

11 7s+¹⁸₇

s²−4+

3 7s+³₇

s²+3 −

(

⁴⁷s²^s+−4⁴⁷+

3 7s+³₇ s²+3 − 1

s−1

)

= ^s+2

s²−4+ 1 s−1= 1

s−2+ 1 s−1

×4＋× (s²−1) [^−12+s²⁽^s²⁻¹⁾]^Y₂( s )=8 s +12+(s²−1)(^s+2−s−1⁴ )

(s⁴−s²−12)Y₂(s)=8s+12+s³−s+2s²−2−⁴⁽^s²⁻¹⁾ s−1