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
Mars2501 [29]
3 years ago
9

Consider the initial value problem y′+2y=4t,y(0)=8.

Mathematics
1 answer:
Xelga [282]3 years ago
5 0

Answer:

Please read the complete procedure below:

Step-by-step explanation:

You have the following initial value problem:

y'+2y=4t\\\\y(0)=8

a) The algebraic equation obtain by using the Laplace transform is:

L[y']+2L[y]=4L[t]\\\\L[y']=sY(s)-y(0)\ \ \ \ (1)\\\\L[t]=\frac{1}{s^2}\ \ \ \ \ (2)\\\\

next, you replace (1) and (2):

sY(s)-y(0)+2Y(s)=\frac{4}{s^2}\\\\sY(s)+2Y(s)-8=\frac{4}{s^2}  (this is the algebraic equation)

b)

sY(s)+2Y(s)-8=\frac{4}{s^2}\\\\Y(s)[s+2]=\frac{4}{s^2}+8\\\\Y(s)=\frac{4+8s^2}{s^2(s+2)} (this is the solution for Y(s))

c)

y(t)=L^{-1}Y(s)=L^{-1}[\frac{4}{s^2(s+2)}+\frac{8}{s+2}]\\\\=L^{-1}[\frac{4}{s^2(s+2)}]+L^{-1}[\frac{8}{s+2}]\\\\=L^{-1}[\frac{4}{s^2(s+2)}]+8e^{-2t}

To find the inverse Laplace transform of the first term you use partial fractions:

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

Thus, you have:

y(t)=L^{-1}[\frac{4}{s^2(s+2)}]+8e^{-2t}\\\\y(t)=L^{-1}[\frac{-1}{s}+\frac{2}{s^2}]+L^{-1}[\frac{1}{s+2}]+8e^{-2t}\\\\y(t)=-1+2t+e^{-2t}+8e^{-2t}=-1+2t+9e^{-2t}  

(this is the solution to the differential equation)

You might be interested in
The following set of coordinates represents which figure?(-5,2), (-3,4), (1,0), (-1,-2)
11Alexandr11 [23.1K]
It is B, rectangle. Sort of sideways.
6 0
2 years ago
Read 2 more answers
(1/3)y2 + 12 = 5x where x = 3
IgorLugansk [536]
If x=3
(1/3)y2+12=15
(1/3)y2=15-12
(1/3)y2=3
y2=3*3
y2=9
y=3
8 0
3 years ago
Need help on the 2 questions ASAP and SHOW WORK PLEASE !!!! 50 points ????
zaharov [31]

Answer:

-4 - 8-=-4

Step-by-step explanation:

4 0
2 years ago
What is the fraction and decimal of 95%
zhuklara [117]
.95 in decimal form and 95/100 as a fraction or 19/20

7 0
3 years ago
Read 2 more answers
Use the equation p = 831 where p = pressure and
Inga [223]

Answer:

The lower the volume, the higher the pressure.

Step-by-step explanation:

Volume and pressure are inverse to each other -- so the more pressure, the less volume, and vice versa. Think of it like this: if you were squished into a tiny room, there would be a lot of pressure on you, and not a lot of volume to move in!

You can also think of it numerically. The higher the V in that equation (the higher the denominator) the lower the p in the equation.

6 0
3 years ago
Read 2 more answers
Other questions:
  • What is a rectangle trampoline that is 6 feet long and 12 feet wide what is the area of her Trampoline in square feet
    6·1 answer
  • If F'(x)=<img src="https://tex.z-dn.net/?f=%20%5Csqrt%7B1%2Bx%5E%7B3%7D%20%7D%20" id="TexFormula1" title=" \sqrt{1+x^{3} } " alt
    15·1 answer
  • Which term shows a product of a number and a variable? 5 + 6x - 7 + 4 <br> A. 5 B. 6x C. -7 D. 4
    13·2 answers
  • 2(x+y)/3 for x =6and y=7
    14·2 answers
  • Would you rather have a cube of gold that measures 25 mm on each side, or two cubes of gold , one is 24 mm per side, and one is7
    14·1 answer
  • Which graph represents an increasing function?
    6·1 answer
  • X is less than or equal to-86
    12·1 answer
  • What is 1/4(8+x+4) simplified using the distributive property
    6·2 answers
  • 10. The figure below is a uniform cross-section of a hall 20 m long and 7 m wide.
    12·1 answer
  • Can someone please help me? :(
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!