2nd order DE question :confused:

In summary, the student has attempted to solve an equation for motion with no rubber band using methods learned in previous sections of the course. They have also attempted to solve for initial conditions. However, they are having difficulty understanding what needs to be done to find solutions. They are missing information on how to solve the equation and are having difficulty selecting initial conditions.
  • #1
Kevin2341
57
0

Homework Statement



The equation for the undamped motion with no rubber band:
y" + k1y = -10
k1 = any number between 12 and 13

Find exact solutions using a couple different initial conditions

And then plot this phase plane using some software


The Attempt at a Solution


So I know ahead of time that my solution needs to be in a vector form in order to plot it in a phase plane (using the software that I have for this class)

Here's my attempt thus far:
I have chosen my k1 to be 12.25 (12.25 squared is 3.5, I tried to pick nice numbers, at least as nice as a number between 12 and 13 can be)

Making y' = v
v' = -12.25y -10

Here I am getting really messed up. I know the shortcut where you can convert the y" to a λ2, the y' to a λ, and the y becomes a constant, so basically you get a polynomial equation in which you can find the roots (eigenvalues) and work from there. However, in every system we worked with in this class, there was never in one condition a problem involving a constant, and I can't seem to find any help pertaining as to what I do with it.

My normal course of action here would be to find a corresponding vector A, and from there, using the eigenvalues, I can find the eigenvectors. I know that because of the value of y (being between 12 and 13), that the eigenvalues will be imaginary. This will lead to using euler's formula, and from there it's a matter of selecting initial conditions and graphing this thing, however, I can't figure out what I am supposed to do with that darn 10. Does it just add into 12.25 to become 22.25? Does it just "showup" somewhere later on?

I could really use some help
 
Physics news on Phys.org
  • #2
12.25 squared is 3.5?

It seems there is a portion of the problem statement omitted. What does your DE describe?

You have a non-homogeneous second order ODE. There is a specific procedure which must be followed in order to obtain a solution. Do you remember what this procedure is? Have you selected your initial conditions?
 
  • #3
A Mass-Spring System with a Rubber Band

1. The equation for undamped motion with no rubber band is

y" + k1 = -10

Once you choose a value for k1 (between 12 and 13) use methods learned in sections 3.6 and 4.1 to obtain
exact solutions for a couple of different initial conditions. Use PPlane to show phase portraits for these
solutions and give a verbal description of the motion.

As for the method on solving this: We have been using the "guessing" method. In which you make an educated guess for a value of y, then take the y' and y" of that, and plug it into the equation (It usually comes out to being in a form of Ae^(lambda*t)) And then you solve for A, and plug that A value in for your educated guess giving you a Yh

and for guessing initial conditions, I'll be quite honest in saying I have absolutely no idea where and how to pick initial conditions. I can never figure out if the y and y' values are related to each other, or if they are just arbitrary values picked. It seems a lot of the values I've seen in examples in class are usually something along the lines of:
y(0) = 2 (or some constant number)
y'(0) = 0

which leads me to think that the y' is usually representing (obviously) the derivative of the y initial.
 

1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves a function, its first derivative, and its second derivative. It describes how the function changes over time or space.

2. How is a second order differential equation different from a first order differential equation?

A first order differential equation only involves the function and its first derivative, while a second order differential equation also includes the second derivative. This makes it a more complex equation with potentially different types of solutions.

3. What are some applications of second order differential equations?

Second order differential equations are commonly used in physics, engineering, and other fields to model real-world phenomena such as motion, vibrations, and electrical circuits. They can also be used in financial modeling and population dynamics.

4. How do you solve a second order differential equation?

There are several methods for solving second order differential equations, including separation of variables, substitution, and using a power series. Some equations can also be solved numerically using computer software.

5. What are the initial conditions for a second order differential equation?

The initial conditions for a second order differential equation are typically two values: the initial value of the function and the initial value of its first derivative. These values help determine the specific solution to the equation.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
119
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
194
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
195
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
977
  • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top