Second Order Equations Can Anybody help me? Greatly appreciated

In summary, the conversation is about a homework problem involving a mass on a spring with given initial conditions. The goal is to find the position of the mass at a given time, determine if it crosses the equilibrium position, and find the time and distance when the mass is furthest from equilibrium. The solution involves solving a second order differential equation and checking if it satisfies the boundary conditions.
  • #1
xinerz
3
0
Second Order Equations! Can Anybody help me?? Greatly appreciated!

Homework Statement


Interpret x(t) as the position of a mass on a spring at time t where x(t) satisfies

x'' + 4x' + 3x = 0.
Suppose the mass is pulled out, stretching the spring one unit from its equilibrium position, and given an initial velocity of +2 units per second.

(A) Find the position of the mass at time t.
(B) Determine whether or not the mass ever crosses the equilibrium position of x = 0.
(C) When (at what time) is the mass furthest from its equilibrium position? Approximately how far from the equilibrium position does it get?


Homework Equations


Previous problems on this homework set include transforming the initial value problem into a solution that looks partially like the following:

(example):
y = (1/3)e^(-4t) + (2/3)e^(-4t)



The Attempt at a Solution



I've attempted the following:
x" + 4x' + 3x = 0 --> r^2 + 4r + 3 = 0, solved for r

r = -3 or -1
therefore y=e^(-3t) , y=e^(-t)
y(t) = Ae^(-3t) + Be^(-t)

solved for A and B both = -1/2

however, I'm not sure that this is right.

THANK YOU!
 
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  • #2
Does it satisfy the differential equation and boundary conditions? If so, it's probably right.
 
  • #3
so to find the position at t, i just solve for y in terms of t? like
t = something

also, how would i show if the mass crossed equilibrium at x = 0?
 
  • #4
No, y(t) is the position at t. And you seem to be using x and y to refer to the same thing. Try graphing the function to see if it crosses 0.
 
  • #5
Or just set it equal to zero and see if there is a solution.
 
  • #6
thanks! i got it you guys :)
thanks for all the help!
HAPPY NEW YEAR!
 

1. What are second order equations?

Second order equations are differential equations that involve second derivatives. They are commonly used to model physical systems that involve acceleration, such as motion and vibrations.

2. What is the general form of a second order equation?

The general form of a second order equation is y'' + p(x)y' + q(x)y = g(x), where p(x) and q(x) are functions of x, and g(x) is a forcing function.

3. What is the order of a second order equation?

The order of a second order equation is 2, because it involves second derivatives.

4. How do you solve a second order equation?

There are several methods for solving second order equations, including the method of undetermined coefficients, the variation of parameters method, and the Laplace transform method. The specific method used depends on the form of the equation and the initial conditions.

5. What are some real-life applications of second order equations?

Second order equations have many applications in physics, engineering, and other fields. They are commonly used to model the motion of objects under the influence of forces, such as pendulums and springs. They are also used in electrical engineering to describe the behavior of circuits and in chemical engineering to model reaction rates.

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