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
olchik [2.2K]
3 years ago
14

Consider the system of differential equations dxdt=−4ydydt=−4x. Convert this system to a second order differential equation in y

by differentiating the second equation with respect to t and substituting for x from the first equation. Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0)=4 and y(0)=5, what are x and y?
Mathematics
1 answer:
koban [17]3 years ago
7 0

\dfrac{\mathrm dy}{\mathrm dt}=-4x\implies x=-\dfrac14\dfrac{\mathrm dy}{\mathrm dt}\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}

Substituting this into the other ODE gives

-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}=-4y\implies y''-16y=0

Since x(t)=-\dfrac14y'(t), it follows that x(0)=-\dfrac14y'(0)=4\implies y'(0)=-16. The ODE in y has characteristic equation

r^2-16=0

with roots r=\pm4, admitting the characteristic solution

y_c=C_1e^{4t}+C_2e^{-4t}

From the initial conditions we get

y(0)=5\implies 5=C_1+C_2

y'(0)=16\implies-16=4C_1-4C_2

\implies C_1=\dfrac12,C_2=\dfrac92

So we have

\boxed{y(t)=\dfrac12e^{4t}+\dfrac92e^{-4t}}

Take the derivative and multiply it by -1/4 to get the solution for x(t):

-\dfrac14y'(t)=\boxed{x(t)=-\dfrac12e^{4t}+\dfrac92e^{-4t}}

You might be interested in
Sally has 56 stamps. Sarah has 3 times as many stamps as Sally. Their brother , Peter has 45 stamps less than Sarah. How many st
aleksandr82 [10.1K]

Answer:

peter will have 123 stamps.

Step-by-step explanation:

This is because 53 times 3 is 159. Sarah has 159 stamps. Subtract 45 from 159 and you will get 123.

7 0
3 years ago
Read 2 more answers
Solve by factoring
True [87]

2×2-13(19÷13)+15=0

So x= 19/13

7 0
3 years ago
PLS HELP I WILL BRAINLIEST 15 PTS
Mnenie [13.5K]

Answer:17 dollars per hour per hour

Step-by-step explanation:

Hope this helps

5 0
3 years ago
The figures in the oval on the left are parallelograms. The figures in the oval on
kozerog [31]

Answer:

The answer is C

Step-by-step explanation:

7 0
2 years ago
HELP ME PLEASE! A representation of a linear equation in two variables is graphed below.
Kamila [148]

Option C, y = 1/2 x -2 is the correct answer

4 0
2 years ago
Other questions:
  • t a hockey game, a vender sold a combined total of 231 sodas and hot dogs. The number of sodas sold was two times the number of
    5·1 answer
  • Iam stuck on this quiz and I need help!!
    5·1 answer
  • What is 9x9 is 81 because if you count by 9 times the 9 is 81
    5·1 answer
  • If seven bags of Takis cost $5.95, how much does one bag cost?
    5·2 answers
  • The circle is centered at the origin and the length of its radius is three. What is the circles equation
    6·1 answer
  • Closing prices of two stocks are recorded for 50 trading days. The sample standard deviation of stock X is 4.665 and the sample
    14·1 answer
  • Hi, thanks to the person that is answering all my questions! the reason im doing this is my teacher said if I finish all my moby
    6·1 answer
  • Given 3(x-2)=6, then x is equal to:<br> Select one:<br> A.5<br> b.4<br> c. 2<br> d.6
    5·1 answer
  • In the xy-plane, the midpoint M of segment AB has coordinates (-1,3), and point A has coordinates
    8·1 answer
  • Three of the following statements are true. Which one is NOT true? <br> |-12| &gt; 1
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!