What is the Behavior of Solutions of a DE with Limits as t Approaches Infinity?

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In summary, the conversation is about finding the general solution of a given differential equation and using it to determine the behavior of solutions as t approaches infinity. After some work, the solution is found to be y=-3et+ce2t, with the limit as t approaches infinity being either positive or negative infinity depending on the value of c. The book states that all solutions will increase exponentially due to the dominance of e2t over et.
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Homework Statement



Find the general solution of the given differential equation, and use it to determine how
solutions behave as t→∞.

y' − 2y = 3et



Homework Equations



DE

The Attempt at a Solution



After some work, I got y=-3et+ce2t . Now I have problems in getting the limit as t goes to infinity. C can possibly be a positive or negative value. In case it is -ve, the answer goes to negative infinity. If it is positive, I can't really figure out what would the limit be. In the book however, it is written that 'It follows
that all solutions will increase exponentially'. HOW?
 
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e2t will beat et, so it will go to positive or negative infinity according to the sign of c.
 

FAQ: What is the Behavior of Solutions of a DE with Limits as t Approaches Infinity?

1. What is a DE problem involving limits?

A DE (differential equation) problem involving limits is a mathematical problem that involves finding a solution to a differential equation while also considering the behavior of the solution as the independent variable approaches a certain value, known as the limit.

2. How are limits used in solving DE problems?

Limits are used to determine the behavior of the solution to a DE problem as the independent variable approaches a certain value. This helps to narrow down the possible solutions and find a specific solution that satisfies both the DE and the given limit.

3. What is the importance of considering limits in DE problems?

Considering limits in DE problems is important because it helps to ensure that the solution is valid and satisfies all the given conditions. It also helps to determine the uniqueness of the solution and provides a more complete understanding of the behavior of the solution.

4. Are there any limitations to using limits in DE problems?

Yes, there are limitations to using limits in DE problems. One limitation is that not all DE problems can be solved using limits, as some may have multiple solutions or no solution at all. Additionally, limits may not always provide a unique solution, so other techniques may need to be used in conjunction with limits.

5. What are some common methods used to solve DE problems involving limits?

Some common methods used to solve DE problems involving limits include substitution, separation of variables, and using special functions such as Laplace transforms. These methods help to simplify the DE and make it easier to apply the given limits to find a solution.

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