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kondor19780726 [428]
4 years ago
11

Consider the initial value problem

Mathematics
1 answer:
uysha [10]4 years ago
8 0

Answer with Step-by-step explanation:

We are given that initial value problem

a.4u''-u'+2u=0

u(0)=3,u'(0)=0

Auxillary equation

4m^2-m+2=0

m=\frac{1\pm\sqrt{(-1)^2-4(4)(2)}}{2(4)}

By using quadratic formula:x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

m=\frac{1\pm\sqrt{1-32}}{8}

m=\frac{1\pm i\sqrt{31}}{8}=\frac{1}{8}\pm i\frac{\sqrt{31}}{8}

Therefore, the general solution of the initial value problem

u(t)=e^{\frac{t}{8}}(c_1cos\frac{\sqrt{31}}{8}t+c_2sin\frac{\sqrt{31}}{8}t)

Substitute t=0 and u(0)=3

3=c_1

Differentiate w.r.t t

u'(t)=\frac{1}{8}e^{\frac{t}{8}}(c_1cos\frac{\sqrt{31}}{8}t+c_2sin\frac{\sqrt{31}}{8}t)+e^{\frac{t}{8}}(-\frac{\sqrt{31}}{8}c_1sin\frac{\sqrt{31}}{8}t+\frac{\sqrt{31}}{8}c_2cos\frac{\sqrt{31}}{8}t)

Substitute t=0 and u'(0)=0

0=\frac{1}{8}c_1+\frac{\sqrt{31}}{8}c_2

Substitute the value of c_1

0=\frac{1}{8}\times 3+\frac{\sqrt{31}}{8}c_2

-\frac{3}{8}=\frac{\sqrt{31}}{8}c_2

c_2=-\frac{3}{8}\times \frac{8}{\sqrt{31}}

c_2=-\frac{3}{\sqrt{31}}

Substitute the values

u(t)=e^{\frac{t}{8}}(3cos\frac{\sqrt{31}}{8}t-\frac{3}{\sqrt{31}}sin\frac{\sqrt{31}}{8}t)

(b) For t>0 \mid u(t)\mid=10

We cannot find the solution of  \mid u(t)\mid=10analytically

But we can use the graph to find the approximate value of the solution.

By graph , the approximate value of t=12.2 for which  \mid u(t)\mid=10

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100 pts if u do it right!
Reil [10]
1.) 4x+8
——
5x-5

2) use Photomath because I’m confused

3) 4x^2+x-6
—————
X^2 + 3x
7 0
3 years ago
Which one doesn't belong, and why?<br> 1/2,5/3,2/10,2/5
Sav [38]

Answer:

5/3

Step-by-step explanation:

5/3 doesn't belong because each denominator for the other numbers are all factors of 10 (I think thats how you say it)

Hope this helps

3 0
3 years ago
Consider the equation
worty [1.4K]

Answer:

Step-by-step explanation:

5/3v + 4 + 1/3v= 8.

Multiply each term by 3v

3v(5/3v) + 3v(4) + 3v(1/3v) = 3v(8)

5 + 12v + 1 = 24v

Collecting like terms

12v - 24v = -5 - 1

- 12v = -6

v = -6/-12

v = 1/2

6 0
3 years ago
Mr. Marcucci received a bonus of $496 from his employer. He has to pay 33% of his bonus to taxes. How much will Mr. Marcucci pay
tiny-mole [99]
Edit: 33% is not \frac{1}{3}, therefore my solution is wrong. The correct answer is 0.33 x 496, which is $163.68

The following is the original solution, which is incorrect. 
33% = \frac{1}{3}. Multiply the bonus of $496 by \frac{1}{3} to get the solution of \frac{496}{3}, or 165 and \frac{1}{3}

Therefore the solution is $165.33


7 0
4 years ago
Read 2 more answers
If 3x+8=3x+8 is it one solution or no solution
Nadusha1986 [10]

Answer:

infinite solutions

Step-by-step explanation:

3x+8=3x+8

Subtract 3x from each side

3x-3x+8=3x-3x+8

8=8

Since this is always true, we have infinite solutions


4 0
3 years ago
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