Can the Laplace equation be used to solve for y⁴?

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In summary, this person was asking for help with finding the Laplace transform of the nth derivative, but they don't know how to start. They were also confused about why the Laplace transform would be useful in solving equations involving y^4. They were eventually able to figure out that y'' wasn't the correct variable to solve for, and that the problem had a typo.
  • #1
avi89
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hi , i was asked to solved this but i have no idea how to laplace y^4, can anyone please help?

its the question in the middle

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  • #2
Do you know the general formula for the Laplace transform of the nth derivative?
 
  • #3
avi89 said:
hi , i was asked to solved this but i have no idea how to laplace y^4, can anyone please help?

its the question in the middle

View attachment 84987
If you look at a table of Laplace transforms, like the one attached to this article:

https://en.wikipedia.org/?title=Laplace_transform

You'll see a formula for finding the first or second derivative, f' and f", for example. Look below to the entry for "general differentiation" and f(n), and you'll see a formula for LT for finding higher-order derivatives.

In your case, n = 4, but you appear to be missing some initial conditions necessary to apply the formula, namely y"(0) and y'''(0).
 
  • #4
this is what i was thinking! some initial conditions are missing to use the derivative forumula, so maybe there's another way to solve.
also, notice that this is not the 4th derivative of y, but power(y,4) .. it got me confused
 
  • #5
Why would you use the laplace transform to solve equations involving ## y^4 ## ? You will just get a lot of convolution integrals.
 
  • #6
avi89 said:
this is what i was thinking! some initial conditions are missing to use the derivative forumula, so maybe there's another way to solve.
also, notice that this is not the 4th derivative of y, but power(y,4) .. it got me confused

Are you sure it isn't a badly-written [itex]y''[/itex]? That would make it a question of solving a linear 2nd-order ODE subject to the two given initial conditions.
 
  • #7
aww god, i think you're right!
sighh
 
  • #8
There's a difference between the 4th derivative of y (y'''') and y⁴. Since this latter is a non-linear term and being the Laplace a linear operator, I'm afraid an ODE involving y⁴ has no solution by the Laplace method. It seems to me that your problem has a typographical error.
 

1. What is the Laplace equation?

The Laplace equation is a partial differential equation that describes the distribution of a potential field in space. It is commonly used in physics and engineering to solve problems involving heat transfer, electrostatics, and fluid dynamics.

2. How do I solve the Laplace equation?

There are several methods for solving the Laplace equation, including separation of variables, the method of images, and the Green's function method. The choice of method depends on the specific problem and boundary conditions.

3. What are the applications of the Laplace equation?

The Laplace equation has a wide range of applications in physics and engineering. It is used to model heat flow, electric potential, fluid flow, and other physical phenomena. It is also used in image processing and computer vision applications.

4. What are the boundary conditions for the Laplace equation?

The boundary conditions for the Laplace equation vary depending on the specific problem being solved. However, the most commonly used boundary conditions are the Dirichlet boundary condition, which specifies the value of the potential at a boundary, and the Neumann boundary condition, which specifies the normal derivative of the potential at a boundary.

5. Can the Laplace equation be solved analytically?

Yes, the Laplace equation can be solved analytically using various mathematical techniques. However, in many practical applications, numerical methods are used to approximate the solution due to the complexity of the problem and the boundary conditions.

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