How to Modify Y(t) for Nonhomogenous 2nd Order DE with e^-t and cos(2t) Terms?

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In summary, the conversation discusses a difficult homework problem involving a differential equation and the attempt at finding a solution using a specific formula. The person asking the question suggests modifying the formula to include t, which is confirmed by the other person as the correct approach, despite being a bit tedious.
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jesuslovesu
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Homework Statement


Well I've got another one that totally sucks.
[tex]y'' + 2y' + 5y = 4e^{-t}cos(2t)[/tex]

Homework Equations


The Attempt at a Solution



I tried Y(t) = [tex]Ae^{-t}cos(2t) + Be^{-t}sin(2t)[/tex] but that unfortunately yielded [tex]0 = 4e^{-t} cos(2t)[/tex]

So my question is how does one modify Y(t) in this type of situation? The only thing I can think of is something like [tex]Y(t) = Ae^{-t}t^2cos(2t) + Be^{-t}tsin(2t)[/tex] but that seems rather painful
 
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  • #2
Try,
[tex]
Y(t) = Ae^{-t}tcos(2t) + Be^{-t}tsin(2t)
[/tex]
which is what you wrote down but I changed a t^2 to a t. Yeah, it's kind of painful, but it will work. Without the t's it just the homogeneous solution. You knew that would give you zero, right?
 
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FAQ: How to Modify Y(t) for Nonhomogenous 2nd Order DE with e^-t and cos(2t) Terms?

1. What is a nonhomogenous 2nd order differential equation?

A nonhomogenous 2nd order differential equation is a mathematical equation that involves the second derivative of a function, as well as other terms that are not related to the function itself. These additional terms are known as the nonhomogenous or inhomogenous part of the equation.

2. How is a nonhomogenous 2nd order DE different from a homogenous 2nd order DE?

A homogenous 2nd order differential equation only contains terms that are related to the function itself, while a nonhomogenous 2nd order DE has additional terms that are not related to the function. This makes solving nonhomogenous equations more complex, as the solution involves both the general solution of the homogenous equation and a particular solution for the nonhomogenous part.

3. What are some real-world applications of nonhomogenous 2nd order DEs?

Nonhomogenous 2nd order DEs are used in various fields of science and engineering, including physics, biology, and economics. They can be used to model the behavior of systems that are affected by external forces or inputs, such as the motion of a pendulum under the influence of air resistance, or the growth of a population with varying birth and death rates.

4. How do you solve a nonhomogenous 2nd order DE?

To solve a nonhomogenous 2nd order DE, you first need to find the general solution of the corresponding homogenous equation. Then, you need to find a particular solution for the nonhomogenous part using methods such as undetermined coefficients or variation of parameters. Finally, you combine the general solution and the particular solution to obtain the complete solution to the nonhomogenous equation.

5. Are there any specific techniques for solving nonhomogenous 2nd order DEs?

Yes, there are several specific techniques for solving nonhomogenous 2nd order DEs, such as the method of undetermined coefficients, the method of variation of parameters, and the Laplace transform method. Each of these methods has its own advantages and limitations, and the choice of which one to use depends on the form of the nonhomogenous equation and the initial conditions given.

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