Question on Ordinary Differential Eqn

In summary, the displacement of a driven mass-spring system is described by the differential equation 3y" + 14y =7cos(2t) with initial value conditions y(0) = 0, y' (0) = 0. To determine if the system is damped or un-damped, one must look at the discriminant of the quadratic equation obtained from using undetermined coefficients to solve the homogeneous equation. If the discriminant is negative, the system is under-damped. If it is positive, it is over-damped. If it is zero, it is critically damped. Resonance occurs when the forcing frequency and the natural frequency of the system are the same. The solution to the
  • #1
sero2000
27
0
The displacement y(t) of a driven mass-spring system is described by the differential equation 3y" + 14y =7cos(2t) with initial value conditions y(0) = 0, y' (0) = 0

a.) is this system damped or un-damped?
b.) Is this system resonant?
c.) Write the solution to the IVP in terms of a product of 2 sine functions
d.) What is the frequency of the beats

Not too sure how to start on this question. Any help to jump start the thinking process would be really appreciated :D.
 
Physics news on Phys.org
  • #2
An obvious way to start would be to find the solution to the differential equation! Have you done that?

Do you know what "damped", "un-damped", and "resonant" mean?
 
  • #3
Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?
 
Last edited:
  • #4
sero2000 said:
Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?

That's correct for the general solution of the DE. But now you need to satisfy the initial conditions to get the specific solution to your problem. Then you will be ready for what's next.
 
  • #5
sero2000 said:
Yup i managed to get the general solution!

y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

Still can't find what is resonant though. does it have something to do with the sine waves?

Is resonant another word for critically damped?

Either way, if you use undetermined coefficients initially to determine the solution to your homogeneous equation, the way to tell its state of damping it to look at the discriminant for your quadratic equation.

If it's negative, it's under-damped.

If it's positive, it's over-damped.

If it's zero, it's critically damped.
 
  • #6
sero2000 said:
Still can't find what is resonant though. does it have something to do with the sine waves?

It has to do with the relationship between the forcing frequency and the natural frequency. Here's a link I think you will find very helpful, although I would expect similar information to be in your text:

http://www.marietta.edu/~mmm002/Math302/Lectures/Ch4Sec3.pdf
 
Last edited by a moderator:
  • #7
Rellek said:
Is resonant another word for critically damped?

No. See the link in post #6.
 

FAQ: Question on Ordinary Differential Eqn

What is an ordinary differential equation (ODE)?

An ordinary differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves only one independent variable and is used to model many real-world phenomena, such as motion, population growth, and chemical reactions.

What is the difference between an ODE and a partial differential equation (PDE)?

The main difference between an ODE and a PDE is the number of independent variables. An ODE involves only one independent variable, while a PDE involves more than one. Additionally, the derivatives in an ODE are with respect to a single variable, while in a PDE they can be with respect to multiple variables.

How are ODEs solved?

There are various methods for solving ODEs, including analytical, numerical, and graphical methods. Analytical methods involve finding an exact solution to the equation, while numerical methods involve approximating the solution using algorithms. Graphical methods use visual representations, such as direction fields and phase portraits, to understand the behavior of the solution.

What are the applications of ODEs?

ODEs are used in many fields, including physics, engineering, economics, and biology. They are used to model and predict the behavior of systems that change over time, such as the motion of objects, the growth of populations, and the spread of diseases. ODEs are also essential in the development of mathematical models for understanding complex phenomena.

Can ODEs have multiple solutions?

Yes, ODEs can have multiple solutions, although not all ODEs do. In some cases, an ODE may have a family of solutions that differ by a constant, known as the general solution. In other cases, there may be multiple distinct solutions, known as particular solutions, that satisfy the ODE for specific initial conditions. The existence and uniqueness of solutions depend on the properties of the ODE and the initial conditions.

Similar threads

Given unit impulse response, obtain differential equation
Replies
7
Views
973
Avoid unpleasant integrals in solving IVP
Replies
3
Views
1K
ODE: System of Linear Equations usuing Diff. Operator
Replies
4
Views
1K
Differential equation for a vibration problem
Replies
1
Views
1K
Differential Equations: linearity principle
Replies
4
Views
2K
MHB Displacement of mass spring system
Replies
1
Views
2K
Ordinary differential equations
Replies
2
Views
1K
Find the differential equation or system of differential equations
Replies
5
Views
2K
Perturbation Methods: Asymptotic Matching Question
Replies
2
Views
1K
I Verifying properties of Green's function
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top