1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Mkey [24]
3 years ago
14

(5) Find the Laplace transform of the following time functions: (a) f(t) = 20.5 + 10t + t 2 + δ(t), where δ(t) is the unit impul

se function (b) f(t) = e −t + 4e −4t + te−3t
Mathematics
1 answer:
Aloiza [94]3 years ago
3 0

Answer

(a) F(s) = \frac{20.5}{s} - \frac{10}{s^2} - \frac{2}{s^3}

(b) F(s) = \frac{-1}{s + 1} - \frac{4}{s + 4} - \frac{4}{9(s + 1)^2}

Step-by-step explanation:

(a) f(t) = 20.5 + 10t + t^2 + δ(t)

where δ(t) = unit impulse function

The Laplace transform of function f(t) is given as:

F(s) = \int\limits^a_0 f(s)e^{-st} \, dt

where a = ∞

=>  F(s) = \int\limits^a_0 {(20.5 + 10t + t^2 + d(t))e^{-st} \, dt

where d(t) = δ(t)

=> F(s) = \int\limits^a_0 {(20.5e^{-st} + 10te^{-st} + t^2e^{-st} + d(t)e^{-st}) \, dt

Integrating, we have:

=> F(s) = (20.5\frac{e^{-st}}{s} - 10\frac{(t + 1)e^{-st}}{s^2} - \frac{(st(st + 2) + 2)e^{-st}}{s^3}  )\left \{ {{a} \atop {0}} \right.

Inputting the boundary conditions t = a = ∞, t = 0:

F(s) = \frac{20.5}{s} - \frac{10}{s^2} - \frac{2}{s^3}

(b) f(t) = e^{-t} + 4e^{-4t} + te^{-3t}

The Laplace transform of function f(t) is given as:

F(s) = \int\limits^a_0 (e^{-t} + 4e^{-4t} + te^{-3t} )e^{-st} \, dt

F(s) = \int\limits^a_0 (e^{-t}e^{-st} + 4e^{-4t}e^{-st} + te^{-3t}e^{-st} ) \, dt

F(s) = \int\limits^a_0 (e^{-t(1 + s)} + 4e^{-t(4 + s)} + te^{-t(3 + s)} ) \, dt

Integrating, we have:

F(s) = [\frac{-e^{-(s + 1)t}} {s + 1} - \frac{4e^{-(s + 4)}}{s + 4} - \frac{(3(s + 1)t + 1)e^{-3(s + 1)t})}{9(s + 1)^2}] \left \{ {{a} \atop {0}} \right.

Inputting the boundary condition, t = a = ∞, t = 0:

F(s) = \frac{-1}{s + 1} - \frac{4}{s + 4} - \frac{4}{9(s + 1)^2}

You might be interested in
Find the volume of a cone with a 3 cm diameter and a 5 cm height
Darya [45]

Step-by-step explanation:

Substitute r = 3 and h = 5 into the formula. Volume V=13⋅π⋅9⋅5=15πcm3=47.1cm3.

3 0
2 years ago
Answer to -13.62-(27.9)
PolarNik [594]

Answer:

−  1049

Step-by-step explanation:

-13.62-(27.9)

Primero los paréntesis:

-13.62-243

Luego continuamos por las multiplicaciones:

-806 - 243

Finalmente obtenemos:

−  1049

Espero te ayude :)

8 0
3 years ago
Expanded num for 373698
ivolga24 [154]
300000+70000+3000+600+90+8
4 0
2 years ago
Read 2 more answers
Andrea and her friends love cake.Andrea has two cakes.each of her friends are going to eat 2/3 of a cake.How many servings of ca
bogdanovich [222]

Answer:

1/3 of each cake

Step-by-step explanation:

2/3 is more then 1/2

4 0
1 year ago
Read 2 more answers
What would the answer to this question be?<br> A. 40°<br> B. 60°<br> C. 80°<br> D. 100°
Alex777 [14]

Answer:

D. 100°

Step-by-step explanation:

A straight line is 180°. Add 50° and 30° (which is 80°) and subtract the total from 180°.

50 + 30 = 80

180 - 80 = 100

8 0
3 years ago
Read 2 more answers
Other questions:
  • Function f is a cosine function with period 8π, amplitude 3, and a local maximum at f(0) = –3. What is the equation of the midli
    11·2 answers
  • One store sells milk for $3.00 per gallon. A second store sells milk for $0.80 per quart. If you need two gallons, which store s
    6·2 answers
  • A pizza place sells pizzas for $7 each plus a $4 delivery charge per order. If Pat orders 3 pizzas to be delivered, what will be
    15·2 answers
  • Write $<br> 55.26 in expanded form
    13·2 answers
  • Please help fasttttttt
    11·1 answer
  • I need to know the surface area. The formula is Area Of Base + Area of lateral faces
    7·1 answer
  • Find the volume of a right circular cone that has a height of 4.9 cm and a base with a circumference of 8.4 cm. Round your answe
    12·1 answer
  • What is the area of a square where each side measures 9.4 ft
    9·1 answer
  • Write an equation to show how to calculate the cost of using this company​
    9·1 answer
  • There are 80 people in Ms. Peterson's classes. One-fourth of the students are going on a field trip. Of those students, one-half
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!