Can the Laplace equation be used to solve for y⁴?

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SUMMARY

The discussion centers on the application of the Laplace transform to the expression y⁴. Participants clarify that y⁴ represents a nonlinear term, while the Laplace transform is a linear operator. Consequently, using the Laplace transform to solve equations involving y⁴ is inappropriate, as it leads to convoluted integrals. Additionally, the need for initial conditions, such as y"(0) and y'''(0), is emphasized for applying the Laplace transform to higher-order derivatives.

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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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Do you know the general formula for the Laplace transform of the nth derivative?
 
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).
 
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
 
Why would you use the laplace transform to solve equations involving ## y^4 ## ? You will just get a lot of convolution integrals.
 
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 y''? That would make it a question of solving a linear 2nd-order ODE subject to the two given initial conditions.
 
aww god, i think you're right!
sighh
 
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.
 

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