Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

S

By inspection, find a one parameter family of solutions of the differential equation xy' = 4y. Verify that each member of its family also satisfies the initial condition y(0) = 0

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. imranmeah91
    Best Response
    You've already chosen the best response.
    Medals 0

    x y' -4y=0 y' - 4/x y=0 now we can better inspect

    • 2 years ago
  2. S
    Best Response
    You've already chosen the best response.
    Medals 1

    Ok thanks i get it so far, but I'm confused by the wording.. one parameter family.. i'm not sure where to go from there

    • 2 years ago
  3. imranmeah91
    Best Response
    You've already chosen the best response.
    Medals 0

    since it say inspection, I don't think we have to try ti solve it

    • 2 years ago
  4. Jemurray3
    Best Response
    You've already chosen the best response.
    Medals 1

    When they say by inspection, they mean just try to see through the problem to the solution without any serious work. It can be done. Note: \[ x y' - 4y = 0 \rightarrow xy' = 4y\] So taking the derivative of the function and multiplying by x is the same as multiplying the function by 4. To me, this suggests something of the form \[ y = A x^4\] where a is an arbitrary constant.

    • 2 years ago
  5. Jemurray3
    Best Response
    You've already chosen the best response.
    Medals 1

    A*

    • 2 years ago
  6. S
    Best Response
    You've already chosen the best response.
    Medals 1

    OOo i think i get it now thank you!

    • 2 years ago
  7. Jemurray3
    Best Response
    You've already chosen the best response.
    Medals 1

    The full solution, in contrast, is the following: \[ xy' - 4y = 0\] \[ y' - \frac{4}{x} y = 0\] Introduce an integrating factor \[e^{\mu(x)} \] so \[[ye^{\mu(x)} ]' = y'e^\mu + \mu'ye^\mu\] Identifying \[ \mu'e^\mu = -\frac{4}{x}e^\mu \rightarrow \mu = e^{-4 \ln(x)} = e^{\ln(x^{-4})} = x^{-4} \] So multiplying through and grouping together, \[ [x^{-4}y]' = 0\] so \[ x^{-4}y = A\] and finally \[y =Ax^4\] That would be showing all the rigorous work... what we did is called by inspection, or just looking at it ;)

    • 2 years ago
  8. S
    Best Response
    You've already chosen the best response.
    Medals 1

    Alright that explains it, thank you!

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.