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
Anna11 [10]
3 years ago
9

Y''+y'+y=0, y(0)=1, y'(0)=0

Mathematics
1 answer:
mars1129 [50]3 years ago
3 0

Answer:

y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+\frac{1}{\sqrt{3}}\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )

Step-by-step explanation:

A second order linear , homogeneous ordinary differential equation has form ay''+by'+cy=0.

Given: y''+y'+y=0

Let y=e^{rt} be it's solution.

We get,

\left ( r^2+r+1 \right )e^{rt}=0

Since e^{rt}\neq 0, r^2+r+1=0

{ we know that for equation ax^2+bx+c=0, roots are of form x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} }

We get,

y=\frac{-1\pm \sqrt{1^2-4}}{2}=\frac{-1\pm \sqrt{3}i}{2}

For two complex roots r_1=\alpha +i\beta \,,\,r_2=\alpha -i\beta, the general solution is of form y=e^{\alpha t}\left ( c_1\cos \beta t+c_2\sin \beta t \right )

i.e y=e^{\frac{-t}{2}}\left ( c_1\cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )

Applying conditions y(0)=1 on e^{\frac{-t}{2}}\left ( c_1\cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right ), c_1=1

So, equation becomes y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )

On differentiating with respect to t, we get

y'=\frac{-1}{2}e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )+e^{\frac{-t}{2}}\left ( \frac{-\sqrt{3}}{2} \sin \left ( \frac{\sqrt{3}t}{2} \right )+c_2\frac{\sqrt{3}}{2}\cos\left ( \frac{\sqrt{3}t}{2} \right )\right )

Applying condition: y'(0)=0, we get 0=\frac{-1}{2}+\frac{\sqrt{3}}{2}c_2\Rightarrow c_2=\frac{1}{\sqrt{3}}

Therefore,

y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+\frac{1}{\sqrt{3}}\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )

You might be interested in
Please help steps thx
ExtremeBDS [4]

Answer:

240 cubic metres

Step-by-step explanation:

I'm not really sure for Q17 but I do know Q18.

Q18. Volume of cuboid = Length×Breadth×Height

=12×4×5

=240 cubic metres

hope it helps!!!

8 0
2 years ago
<RQT is a straight angle. What are m<RQS and m<TQS​
never [62]

Answers:

  • x = 11
  • angle RQS = 106 degrees
  • angle SQT = 74 degrees

===========================================================

Explanation:

Straight angles are always 180 degrees in measure.

The two smaller angles shown add up to 180

(angle RQS) + (angle SQT) = angle RQT

(9x+7) + (6x+8) = 180

(9x+6x) + (7+8) = 180

15x+15 = 180

15x = 180-15

15x = 165

x = 165/15

x = 11

From here, we then know that,

  • angle RQS = 9x+7 = 9*11+7 = 99+7 = 106 degrees
  • angle SQT = 6x+8 = 6*11+8 = 66+8 = 74 degrees

Note how the two results add to 106+74 = 180 to help confirm the answers.

8 0
3 years ago
Help me please with #12
Helga [31]

Answer:

Am am not that big brain

Step-by-step explanation:

6 0
3 years ago
What is the answer for 0.9 divided by 10 to the power of 3
velikii [3]

Answer:

0.0009

Step-by-step explanation:

0.9 / 10^3 = 0.0009

5 0
3 years ago
Christine is 8 years younger than yuri. Express christine's age in terms of yuri's age
nikitadnepr [17]
Christine is “x” and yuri is “y”

y = x - 8
8 0
3 years ago
Other questions:
  • What is the range of f(x)?
    10·2 answers
  • What is the coefficient of x7 in the expansion of (1 + x)11?
    15·1 answer
  • How to find angles in a triangle
    14·1 answer
  • Asap Algebra 2 hon explanation would be greatly appreciated.
    7·1 answer
  • Given f(x)=x^2 - 6x + 8 and g(x) = x - 2, solve f(x) = g(x) using a table of values.
    10·1 answer
  • For f(x) = 4x+1 and g(x)=x^2-5 find (f + g)(x).<br>​
    8·1 answer
  • What is the approximate scale factor dilating from the blue image to the red image? Explain how you figured this out.
    12·1 answer
  • Lo ran 6 ¼ miles Thelma ran 1 2/3 miles.<br> How many times further did Lo run than Thelma?
    11·1 answer
  • Why is the distributive property so important
    6·1 answer
  • This may be a silly question but what is 21 divided by 24?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!