Having trouble with Laplace Transform for DiffEQ

In summary, The speaker is having difficulty solving a DiffEQ with a summation and is seeking strategies for solving it. They are also not sure if they have correctly typed out the problem because their graphics have not been generated yet.
  • #1
Theelectricchild
260
0
Hello everyone, well thus far in our introduction to Laplace Transforms I am understanding much of what is being shown, however I am having the unsatisfying task of having to solve the following DiffEQ,

[tex]y^3-8y=\sum_{k=0}^{3}\delta(2t-k\pi), y(0)=0, y'(0), y"(0)=0[/tex]

I am having a great difficulty solving this and am overall not really understanding what do with the summation, does anyone have any strategy? Any help is greatly appreciated thank you.
 
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  • #2
Interesting... unfortunately my LaTeX graphics have not been generated yet so I can't even tell if I typed out the problem correctly--- if you can't read the problem disregard for now and ill fix it later, thanks a lot.
 
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  • #3


Hello there, solving differential equations using Laplace transforms can definitely be tricky at first. It's important to have a solid understanding of the basics before tackling more complex problems like the one you mentioned. Here are a few tips that may help you with this particular problem:

1. Start by taking the Laplace transform of both sides of the equation. This will involve using the properties of the Laplace transform, such as linearity and differentiation.

2. The summation term in the equation represents a series of delta functions, which can be converted into a single Laplace transform using the sifting property. This will help simplify the equation.

3. Keep in mind that the Laplace transform of a derivative is equal to s times the Laplace transform of the original function. This will come in handy when solving for y'(0) and y''(0).

4. Once you have the transformed equation, you can use inverse Laplace transforms to find the solution for y(t). This may involve using partial fraction decomposition or other techniques depending on the complexity of the transformed equation.

5. Don't forget to apply the initial conditions given in the problem (y(0)=0, y'(0)=0, y''(0)=0) to find the values of any constants that may arise in the solution.

I hope these tips help you in solving the problem. If you're still having trouble, don't hesitate to reach out to your instructor or classmates for further assistance. Keep practicing and you'll soon master Laplace transforms for differential equations. Good luck!
 

1. What is a Laplace Transform?

A Laplace Transform is a mathematical tool used to solve differential equations in the field of mathematics and engineering. It converts a function of time into a function of a complex variable, which allows for easier manipulation and analysis.

2. Why is the Laplace Transform useful for solving differential equations?

The Laplace Transform is useful because it allows for the conversion of a differential equation into an algebraic equation, which can be easier to solve. It also has properties that make it easier to solve certain types of differential equations, such as initial value problems.

3. What are the key steps for using Laplace Transform to solve a differential equation?

The key steps for using Laplace Transform to solve a differential equation are: 1) taking the Laplace Transform of the differential equation, 2) solving for the transformed function, 3) taking the inverse Laplace Transform to obtain the solution in terms of time, and 4) applying any necessary initial conditions.

4. What are some common challenges when using Laplace Transform to solve differential equations?

Some common challenges when using Laplace Transform include: 1) ensuring the function is in the correct form for the transform, 2) correctly applying the properties of the transform, 3) choosing the appropriate initial conditions, and 4) dealing with complex numbers in the solution.

5. How can I improve my understanding and proficiency with Laplace Transform for differential equations?

The best way to improve your understanding and proficiency with Laplace Transform is through practice. Work through various examples and problems, and make sure to check your work. It can also be helpful to study the properties and applications of the transform and to seek help from a tutor or teacher if needed.

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