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
scoray [572]
3 years ago
14

Given the second order homogeneous constant coefficient equation y′′−4y′−12y=0 1) the characteristic polynomial ar2+br+c is r^2-

4r-12 . 2) The roots of auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). 4) Given the initial conditions y(0)=1 and y′(0)=−26 find the unique solution to the IVP y= .
Mathematics
1 answer:
Vitek1552 [10]3 years ago
7 0

Answer:

The initial value problem y(x) = 4 e^{-2x} -3 e^{6 x}

Step-by-step explanation:

<u>Step1:</u>-

a) Given second order homogenous constant co-efficient equation

y^{ll} - 4y^{l}-12y=0

Given equation in the operator form is (D^{2} -4D-12)y=0

<u>Step 2</u>:-

b) Let f(D) = (D^{2} -4D-12)y

Then the auxiliary equation is (m^{2} -4m-12)=0

Find the factors of the auxiliary equation is

m^{2} -6m+2m-12=0

m(m-6) + 2(m-6) =0

m+2 =0 and m-6=0

m=-2 and m=6

The roots are real and different

The general solution y = c_{1} e^{-m_{1} x} + c_{2} e^{m_{2} x}

the roots are m_{1} = -2 and m_{2} = 6

The general solution of given differential equation is

y = c_{1} e^{-2x} + c_{2} e^{6 x}

<u>Step 3</u>:-

C) Given initial conditions are y(0) =1 and y1 (0) =-26

The general solution of given differential equation is

y(x) = c_{1} e^{-2x} + c_{2} e^{6 x}   .....(1)

substitute x =0 and y(0) =1

y(0) = c_{1} e^{0} + c_{2} e^{0}

1 = c_{1}  + c_{2}    .........(2)

Differentiating equation (1) with respective to 'x'

y^l(x) = -2c_{1} e^{-2x} + 6c_{2} e^{6 x}

substitute x= o and y1 (0) =-26

-26 = -2c_{1} e^{0} + 6c_{2} e^{0}

-2c_{1} + 6c_{2} = -26 .............(3)

solving (2) and (3)  by using substitution method

substitute   c_{2} =1- c_{1} in equation (3)

-2c_{1} + 6(1-c_{1}) = -26

on simplification , we get

-2c_{1} + 6(1)-6c_{1}) = -26

-8c_{1} = -32

dividing by'8' we get c_{1} =4

substitute c_{1} =4 in equation 1 = c_{1}  + c_{2}

so c_{2} = 1-4 = -3

now substitute c_{1} =4 and c_{2} =-3 in general solution

y(x) = c_{1} e^{-2x} + c_{2} e^{6 x}

y(x) = 4 e^{-2x} -3 e^{6 x}

now the initial value problem

y(x) = 4 e^{-2x} -3 e^{6 x}

You might be interested in
What is greater 1.036 1.306
vichka [17]

Answer:

1.306

Step-by-step explanation:

1.3 is bigger than

1.03

6 0
3 years ago
Read 2 more answers
Which expression represents the product of an unknown number and 19?
Galina-37 [17]
I would go with D. Hope that helps :)
7 0
3 years ago
Read 2 more answers
What fraction is £1 of 13p?
Rasek [7]
I believe it would be 13p im not for sure though
4 0
3 years ago
when converting a linear equation into an an augmented matrix which equation form is needed- slope intercept- negative form- gra
IrinaVladis [17]

<span>When 10 grams of copper and 10 grams of iodine are mixed, what is the
theoretical yield of copper I iodide?</span>

2Cu + I2 → 2CuI

<span>30 g15 g7.5 g20 g</span>
3 0
3 years ago
A pipe is leaking at the rate of 32 fluid ounces per minute. Use dimensional analysis to find out how many gallons the pipe is l
lianna [129]

well first think about it, 60 min in an hour, so 32 fluid ounces times 60 = 1920 fluid ounces, from there you can convert fluid ounces to practically anything, for instance, 1920 fluid ounces = 24 cups.

go to this address to find out more.

https://www.google.com/search?q=another+name+for+observations&rls=com.microsoft:en-US:IE-Address&ie=...

4 0
3 years ago
Read 2 more answers
Other questions:
  • A dice is rolled, then a coin is tossed.what is the probability of getting 5 then tails?
    12·2 answers
  • A state becomes good if its people are good justify this statement​
    13·2 answers
  • Find the midpoint of diagonal AC of ABCD
    6·1 answer
  • What is the numbers to this
    15·1 answer
  • Number 10 plz. Plz answer by Tomorrow plz and thx
    11·2 answers
  • a group buys 6 hot dogs and 3 hamburgers for a total of $27.75. the hamburgers cost $1.75 more than a hot dog. what is he cost o
    8·1 answer
  • A train travels at a speed of 65 miles per hour. How long did it take the train to travel 227.5 miles?
    10·1 answer
  • PLS HELP ME PLS IT"S DUE
    15·1 answer
  • Find the solutions to the equation x2 - 15x - 100 = 0.
    15·2 answers
  • 16. Simplify: look at picture<br> please give detailed steps!!
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!