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
Yuri [45]
3 years ago
10

Consider the following differential equation. x^2y' + xy = 3 (a) Show that every member of the family of functions y = (3ln(x) +

C)/x is a solution of the differential equation. (Do this on paper. Your instructor may ask you to turn in this work.) (b) Illustrate part (a) by graphing several members of the family of solutions on a common screen. (Do this on paper. Your instructor may ask you to turn in this work.) (c) Find a solution of the differential equation that satisfies the initial condition y(1) = 3. (Enter the argument of the logarithmic function in parentheses.) y(x) = (d) Find a solution of the differential equation that satisfies the initial condition y(3) = 1. (Enter the argument of the logarithmic function in parentheses.) y(x) =
Mathematics
1 answer:
Veronika [31]3 years ago
6 0

Answer:

Verified

y(x) = \frac{3Ln(x) + 3}{x}

y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{x}

Step-by-step explanation:

Question:-

- We are given the following non-homogeneous ODE as follows:

                           x^2y' +xy = 3

- A general solution to the above ODE is also given as:

                          y = \frac{3Ln(x) + C  }{x}

- We are to prove that every member of the family of curves defined by the above given function ( y ) is indeed a solution to the given ODE.

Solution:-

- To determine the validity of the solution we will first compute the first derivative of the given function ( y ) as follows. Apply the quotient rule.

                          y' = \frac{\frac{d}{dx}( 3Ln(x) + C ) . x - ( 3Ln(x) + C ) . \frac{d}{dx} (x)  }{x^2} \\\\y' = \frac{\frac{3}{x}.x - ( 3Ln(x) + C ).(1)}{x^2} \\\\y' = - \frac{3Ln(x) + C - 3}{x^2}

- Now we will plug in the evaluated first derivative ( y' ) and function ( y ) into the given ODE and prove that right hand side is equal to the left hand side of the equality as follows:

                          -\frac{3Ln(x) + C - 3}{x^2}.x^2 + \frac{3Ln(x) + C}{x}.x = 3\\\\-3Ln(x) - C + 3 + 3Ln(x) + C= 3\\\\3 = 3

- The equality holds true for all values of " C "; hence, the function ( y ) is the general solution to the given ODE.

- To determine the complete solution subjected to the initial conditions y (1) = 3. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

                         y( 1 ) = \frac{3Ln(1) + C }{1} = 3\\\\0 + C = 3, C = 3

- Therefore, the complete solution to the given ODE can be expressed as:

                        y ( x ) = \frac{3Ln(x) + 3 }{x}

- To determine the complete solution subjected to the initial conditions y (3) = 1. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

                         y(3) = \frac{3Ln(3) + C}{3} = 1\\\\y(3) = 3Ln(3) + C = 3\\\\C = 3 - 3Ln(3)

- Therefore, the complete solution to the given ODE can be expressed as:

                        y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{y}

                           

Download docx
You might be interested in
Walter’s history test scores and Janine’s history test scores are shown on the dot plot below.
Usimov [2.4K]
Hiya!

Right now, both of their medians are 86. This means that both will get the same median if a 65 was added.

I believe A.) would be the correct answer, given it's pretty far away from 83 or 84. My second answer would be B.), but I'm not sure.

If you found this especially helpful, I'd appreciate if you'd vote me Brainliest for your answer. I want to be able to assist more users one-on-one! :)
5 0
2 years ago
Read 2 more answers
I don’t know the answer pls help I’ll give you brainliest!! 19 points.
gladu [14]

Answer:

it is A i think i am a nub tho

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Which of the following correctly uses absolute value to show the distance between –60 and 11?
MissTica

Answer:

C) |–60 + 11| = |–49| = –49 units

Step-by-step explanation:

4 0
3 years ago
Suppose the null hypothesis, H0, is: Darrell has enough money in his bank account to purchase a new television. What is the Type
seropon [69]
<h2>Answer with explanation:</h2>

In statistics, The Type II error occurs when the null hypothesis is false, but fails to be rejected.

Given : Suppose the null hypothesis, H_0, is: Darrell has enough money in his bank account to purchase a new television.

Then , Type II error in this scenario will be when the null hypothesis is false, but fails to be rejected.

i.e. Darrell has not enough money in his bank account to purchase a new television but fails to be rejected.

3 0
3 years ago
A set of data has a normal distribution with a mean of 46 and a standard deviation of 7. Find the percent of data within the fol
kvasek [131]

Answer:

50%

the mean (46) marks the halfway point, so over the mean is 50% and under the mean is 50%

7 0
1 year ago
Other questions:
  • Solve this jjejsjssldhdjdjdksjsjsjs
    7·1 answer
  • Can the polynomial below be factored into a perfect square? If not, select the answer that best describes why not.
    12·2 answers
  • Mel drives a bus 39 weeks in a year.
    9·1 answer
  • The square pyramid has a volume of 27 cubic inches. What is the value of X?
    10·1 answer
  • Can somebody please help me
    14·1 answer
  • An easy way to simplify fractions
    15·1 answer
  • Solve the equation 3x+5y=16 for y
    14·2 answers
  • Simplify<br><br> 1/4 (-12+4/3)
    10·1 answer
  • Can someone explain this geometry problem?? Will give Brainliest!!
    15·1 answer
  • Which graph represents the hyperbola of (x ^ 2)/9 + (y ^ 2)/4 = 1
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!