Match second-order differential equations with their y(t) graph

In summary, the conversation is about solving a problem involving a sinusoidal forced equation with various parameters. The speaker is trying to determine which y(t) graph belongs to the equation and is using a complex approach, but the other person suggests using the given parameters to determine the appropriate graph. They also mention a section in the textbook about overdamped, underdamped, and critically damped systems.
  • #1
Dusty912
149
1

Homework Statement


The problem is in the picture. #17
I would have typed it but there are graphs that are needed for solving it. Basically trying to figure out which y(t) graph belongs to one of the sinusoidal forced equations with various parameters.

Homework Equations


euler's formula eit=cost +isint

polar form of complex numbers

The Attempt at a Solution


so I have been complexifying the right hand side, using the guess and check method with a substituted αewit. Once I find α I convert it into its polar form and then change y particular so I can see it's amplitude. This along with 2π/w for the period. Am I going about this the wrong way?
 

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  • #3
Dusty912 said:

Homework Statement


The problem is in the picture. #17
I would have typed it but there are graphs that are needed for solving it. Basically trying to figure out which y(t) graph belongs to one of the sinusoidal forced equations with various parameters.

Homework Equations


euler's formula eit=cost +isint

polar form of complex numbers

The Attempt at a Solution


so I have been complexifying the right hand side, using the guess and check method with a substituted αewit. Once I find α I convert it into its polar form and then change y particular so I can see it's amplitude.
This doesn't seem like a good approach to me. You're given the parameters for six different versions of the DE. For each of the six problems, substitute in the parameters, after which you should be able to tell, at least qualitatively, which of the six graphs is most appropriate. There should be a section in your textbook that talks about systems that are overdamped, underdamped, and critically damped. Some of the graphs definitely fit some or all of these categories.
Dusty912 said:
This along with 2π/w for the period. Am I going about this the wrong way?
 

1. What are second-order differential equations?

Second-order differential equations are mathematical equations that involve the second derivative of a function. They are commonly used to model physical systems and describe how a system changes over time.

2. How do I match a second-order differential equation with its y(t) graph?

To match a second-order differential equation with its y(t) graph, first identify the type of differential equation (e.g. linear, non-linear) and the order (e.g. second-order). Then, you can use techniques such as substitution or separation of variables to solve the equation and obtain an expression for y(t). Finally, plot the graph of y(t) using the expression obtained.

3. Can a second-order differential equation have multiple solutions?

Yes, a second-order differential equation can have multiple solutions. This is because the equation often contains two arbitrary constants that can take on different values, resulting in different solutions.

4. How do I know if a graph represents a solution to a second-order differential equation?

To determine if a graph represents a solution to a second-order differential equation, you can substitute the values of the independent variable (t) and the dependent variable (y) into the equation and check if it satisfies the equation. If it does, then the graph is a solution to the equation.

5. Are there any real-world applications of second-order differential equations?

Yes, there are many real-world applications of second-order differential equations. They are commonly used in fields such as physics, engineering, economics, and biology to model and analyze various systems and phenomena. Examples include the motion of a pendulum, the growth of a population, and the flow of electricity in a circuit.

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