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
madreJ [45]
3 years ago
5

Consider the following. (A computer algebra system is recommended.) y'' + 3y' = 2t4 + t2e−3t + sin 3t (a) Determine a suitable f

orm for Y(t) if the method of undetermined coefficients is to be used. Y(t) = t(A0t4 + A1t3 + A2t2 + A3t + A4) + (B0t2 + B1t + B2)e−t + C sin 3t + D cos 3t Y(t) = t(A1t3 + A2t2 + A3t + A4) + t(B0t3 + B1t2 + B2t + B3)e−3t + C sin 3t + D cos 3t Y(t) = (A0t4 + A1t3 + A2t2 + A3t + A4) + t(B0t2 + B1t + B2)e−3t + C sin t + D cos t Y(t) = t(A0t4 + A1t3 + A2t2 + A3t + A4) + t(B1t + B2)e−3t + C sin 3t + D cos 3t Y(t) = t(A0t4 + A1t3 + A2t2 + A3t + A4) + t(B0t2 + B1t + B2)e−3t + C sin 3t + D cos 3t (b) Use a computer algebra system to find a particular solution of the given equation. Y(t) =
Mathematics
1 answer:
drek231 [11]3 years ago
3 0

First look for the fundamental solutions by solving the homogeneous version of the ODE:

y''+3y'=0

The characteristic equation is

r^2+3r=r(r+3)=0

with roots r=0 and r=-3, giving the two solutions C_1 and C_2e^{-3t}.

For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.

y''+3y'=2t^4

Assume the ansatz solution,

{y_p}=at^5+bt^4+ct^3+dt^2+et

\implies {y_p}'=5at^4+4bt^3+3ct^2+2dt+e

\implies {y_p}''=20at^3+12bt^2+6ct+2d

(You could include a constant term <em>f</em> here, but it would get absorbed by the first solution C_1 anyway.)

Substitute these into the ODE:

(20at^3+12bt^2+6ct+2d)+3(5at^4+4bt^3+3ct^2+2dt+e)=2t^4

15at^4+(20a+12b)t^3+(12b+9c)t^2+(6c+6d)t+(2d+e)=2t^4

\implies\begin{cases}15a=2\\20a+12b=0\\12b+9c=0\\6c+6d=0\\2d+e=0\end{cases}\implies a=\dfrac2{15},b=-\dfrac29,c=\dfrac8{27},d=-\dfrac8{27},e=\dfrac{16}{81}

y''+3y'=t^2e^{-3t}

e^{-3t} is already accounted for, so assume an ansatz of the form

y_p=(at^3+bt^2+ct)e^{-3t}

\implies {y_p}'=(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}

\implies {y_p}''=(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}

Substitute into the ODE:

(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}+3(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}=t^2e^{-3t}

9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c-9at^3+(9a-9b)t^2+(6b-9c)t+3c=t^2

-9at^2+(6a-6b)t+2b-3c=t^2

\implies\begin{cases}-9a=1\\6a-6b=0\\2b-3c=0\end{cases}\implies a=-\dfrac19,b=-\dfrac19,c=-\dfrac2{27}

y''+3y'=\sin(3t)

Assume an ansatz solution

y_p=a\sin(3t)+b\cos(3t)

\implies {y_p}'=3a\cos(3t)-3b\sin(3t)

\implies {y_p}''=-9a\sin(3t)-9b\cos(3t)

Substitute into the ODE:

(-9a\sin(3t)-9b\cos(3t))+3(3a\cos(3t)-3b\sin(3t))=\sin(3t)

(-9a-9b)\sin(3t)+(9a-9b)\cos(3t)=\sin(3t)

\implies\begin{cases}-9a-9b=1\\9a-9b=0\end{cases}\implies a=-\dfrac1{18},b=-\dfrac1{18}

So, the general solution of the original ODE is

y(t)=\dfrac{54t^5 - 90t^4 + 120t^3 - 120t^2 + 80t}{405}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\dfrac{3t^3+3t^2+2t}{27}e^{-3t}-\dfrac{\sin(3t)+\cos(3t)}{18}

You might be interested in
Your sock drawer has two white socks, four brown socks, and two black socks. You randomly pick a sock and put it on your left fo
zmey [24]

Answer:

The probability of this occurring is 1/7

Step-by-step explanation:

Okay, here is a probability question.

From the question, we can identify the following;

Total number of socks = 2 + 4 + 2 = 8 socks

Since we are not given the order in which the socks was worn, we can assume any order.

Let’s say the right leg socks was worn first.

From the question, the socks on the right leg is a brown one.

So now let’s calculate the probability of selecting a brown socks out of the mix

That would be ;

number of brown socks/ total number of socks = 4/8 = 1/2

Now, for the left foot, we are having white sock here.

Kindly recall that since we already have a sock on the right foot, we have 7 socks left to pick from.

Now we need the probability of picking a white sock from the total 7. That would be ; 2/7

So the probability of both action occurring is simply the product of both probabilities and that is ;

1/2 * 2/7 = 1/7

6 0
3 years ago
Simplify the difference.<br> (6w^2 - 5w - 8) – (7w^2 + 4w - 2)
BlackZzzverrR [31]

Answer:

42w^2-20w^2+16

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
Josh borrowed $50,000 from the bank to buy a new truck. The bank is going to charge him an interest rate of 6.9%. If his loan la
Neko [114]

Answer:

<em>The simple interest is $11,212.5 and the equation to determine Josh's simple interest is I = PRT/100</em>

Step-by-step explanation:

The formula for calculating simple interest is expressed as;

SI = Principal * rate * time/100

SI = 50,000 * 6.9*39/100*12

SI = 500*6.9*39/12

SI = 134,550/12

SI = 11,212.5

<em>Hence the simple interest is $11,212.5 and the equation to determine Josh's simple interest is I = PRT/100</em>

P is the amount borrowed

R is the interest rate

T is the time in years

8 0
3 years ago
7th grade math help me please :))
Sav [38]

Answer:

no because the absolute value of -7 is 7 which is greater than the absolute value of 6 which is 6.

8 0
2 years ago
Help, I need #2,3, and 4.
vredina [299]
I can’t read it , it’s very blurry .





try reuploading.
6 0
2 years ago
Other questions:
  • You have a 10kg bag of sand. The mass of a grain of sand is about 10 to the -3 power gram. About how many grains of sand are in
    5·1 answer
  • A recipe for soup calls for 5 2/3 cups of water. If Sharon wants to make four and one half times the recipe, how much water shou
    13·2 answers
  • Is 0.5 bigher than 0.4
    12·2 answers
  • Please, someone help!
    11·1 answer
  • The temperature in city increased from -12 F at the dawn to 7 F in the afternoon. what was the change in temperature in degrees
    9·1 answer
  • PLEASE HELP BRAINLIST PLEASEEEEEEE
    14·1 answer
  • Lucas's computer took 8 minutes to download 30 songs. How many seconds per song did it take his computer to download?
    9·1 answer
  • What is the area of this figure?
    9·1 answer
  • Pls help I’ll brainlest solve it like this
    14·2 answers
  • PLEASE HELP WILL MARK BRAINLIEST
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!