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
Tresset [83]
3 years ago
10

Find the solution of the given initial value problem: y''- y = 0, y(0) = 2, y'(0) = -1/2

Mathematics
1 answer:
igor_vitrenko [27]3 years ago
4 0

Answer:  The required solution of the given IVP is

y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.

Step-by-step explanation:  We are given to find the solution of the following initial value problem :

y^{\prime\prime}-y=0,~~~y(0)=2,~~y^\prime(0)=-\dfrac{1}{2}.

Let y=e^{mx} be an auxiliary solution of the given differential equation.

Then, we have

y^\prime=me^{mx},~~~~~y^{\prime\prime}=m^2e^{mx}.

Substituting these values in the given differential equation, we have

m^2e^{mx}-e^{mx}=0\\\\\Rightarrow (m^2-1)e^{mx}=0\\\\\Rightarrow m^2-1=0~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mx}\neq0]\\\\\Rightarrow m^2=1\\\\\Rightarrow m=\pm1.

So, the general solution of the given equation is

y(x)=Ae^x+Be^{-x}, where A and B are constants.

This gives, after differentiating with respect to x that

y^\prime(x)=Ae^x-Be^{-x}.

The given conditions implies that

y(0)=2\\\\\Rightarrow A+B=2~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)

and

y^\prime(0)=-\dfrac{1}{2}\\\\\\\Rightarrow A-B=-\dfrac{1}{2}~~~~~~~~~~~~~~~~~~~~~~~~(ii)

Adding equations (i) and (ii), we get

2A=2-\dfrac{1}{2}\\\\\\\Rightarrow 2A=\dfrac{3}{2}\\\\\\\Rightarrow A=\dfrac{3}{4}.

From equation (i), we get

\dfrac{3}{4}+B=2\\\\\\\Rightarrow B=2-\dfrac{3}{4}\\\\\\\Rightarrow B=\dfrac{5}{4}.

Substituting the values of A and B in the general solution, we get

y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.

Thus, the required solution of the given IVP is

y(x)=\dfrac{3}{4}e^x+\dfrac{5}{4}e^{-x}.

You might be interested in
algebraically determine the values of h and k to correctly complete the identity stated below : 2 x cubed - 10x squared + 11x -
11Alexandr11 [23.1K]

Answer:

The values for expression is h = - 2 and k = 5

Step-by-step explanation:

Given algebraic expression can be written as :

2 x³ - 10 x² + 11 x - 7 = ( x - 4 ) × ( 2 x² + h x + 3 ) + k

Now opening the bracket

Or, 2 x³ - 10 x² + 11 x - 7 = x × ( 2 x² + h x + 3 ) - 4 × ( 2 x² + h x + 3  ) + k

Or, 2 x³ - 10 x² + 11 x - 7 = 2 x³ + h x² + 3 x - 2 x² - 4 h x - 12  +k

Or , 2 x³ - 10 x² + 11 x - 7 = 2 x³ + ( h - 2 ) x² + ( 3 - 4 h ) x - 12  + k

Now, equating the equation both sides

I.e  - 10 =  ( h - 2 )

Or , h - 2 = - 10

I.e , h = - 10 + 2

∴ h = - 2

Again , 11  = ( 3 - 4 h )

or, 11 = 3 - 4 h

or, 11 - 3 = - 4 h

or, 8 = - 4 h

∴ h = \frac{-8}{4}

I.e h = - 2

Again

- 7 = - 12 + k

Or, k = - 7 + 12

∴  k = 5

Hence The values for expression is h = - 2 and k = 5    . Answer

5 0
2 years ago
In the third quarter of 2001, the median sale price for a single-family
snow_tiger [21]

Answer:

The % increase is 7.179 but see the remark I made below. What I have given and what it could be is a rounding error. Less than 2 dollars error is not much when you are calculating something for someone just to show them how to do it.

Step-by-step explanation:

242,555 * x/100 = 259,970           Multiply by 100

242,555 * x = 259970 * 100

242,555 * x = 25997000            Divide by 242555

x = 24997000/242444

x = 107.179

That's the total increase.

What you want is 7.179%

If you take 7.179% of 242,555 and add it onto 242555 you should get 259,970

7.1797/100 * 242555 = 17413.02

17413.02 + 242555 = 259968.02

Why isn't it exact. I'm out just about 2 whole dollars. The reason is that the % isn't exact. It's only out to 3 decimal places which apparently is not enough. You can go over my numbers if you want more exactness us use something that is rounded to 7.180 if you like.

8 0
3 years ago
Combine the like terms to create an equivalent expression 5z + 3 -3z
Valentin [98]

Answer:

2z + 3

Step-by-step explanation:

5z + -3z is 2

2z + 3

7 0
3 years ago
Read 2 more answers
Brooks used 2/3 of a can of paint to paint the bird feeder. A full can of paint contains 7/8 of a gallon. How much paint did Bro
Cerrena [4.2K]
When Brooks painted more he used 6/12 of what was left(1/2 of what was left)
6 0
3 years ago
HELPPPPPP pleeeeease
dusya [7]

Answer: x = 2 Hope this helps, please consider making me Brainliest.

Step-by-step explanation:

Let's solve for x:

4x + 6 + 2x = 18

4x + 2x = 12

6x = 12

x = 2

Hope this helps, please consider making me Brainliest.

8 0
2 years ago
Read 2 more answers
Other questions:
  • What is the slope of the equation y=5/4x-7/4?<br><br> A. -7<br> B. -7/4<br> C. 5/4<br> D. 5
    10·1 answer
  • A publishing company is going to have 24,000 books printed. There are between 3 and 4 books out of every 3,000 printed that will
    11·1 answer
  • An ostrich that is 108 inches tall is 20 inches taller than 4 times the height of a kiwi. What is the height of the kiwi in inch
    6·1 answer
  • Using the prostate data from the faraway package with lpsa (log prostate specific antigen) as response and lcavol (log cancer vo
    5·1 answer
  • Points Q and P on the coordinate grid below show the positions of two midfield players of a soccer team:
    13·1 answer
  • Solve 0= 2t^3-21t^2+ 40t
    13·1 answer
  • Explain how you could find the measurements of an enlargement if you know the ratio of the measurements of the original to the m
    10·1 answer
  • Please I need help! I'll give brainiest please! And explain how you got your answer please!
    8·2 answers
  • PLEASE ANSWER ASAP
    9·1 answer
  • 1930 to the nearest 1000​
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!