Solve ODE with Fractional Term: Find Solutions

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In summary, the conversation discussed finding solutions for the equation d2u/d2x + 1/2Lu = 0, and how to remove fractions from the second term. It was also mentioned that usually in an ode, the second term contains Y'(dy/dx) or Y (in this case u), but in this equation, it contains L. The speaker suggested making a characteristic equation from 2d2u/d2x + 1/2Lu = 0, which leads to the equation 2λ2 + Lλ = 0.
  • #1
knockout_artist
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Hi,

This is equation I need to find solutions for
d2u/d2x + 1/2Lu = 0 where L(x)

I understand we can remove fraction from second term.
2 [d2 u/d2x ] + Lu = 0

now how do I find solution of this equation ?

How do we deal with L ? because usually we have Y'(dy/dx or in this case du/dx ) or Y (in this case u) in second term in an ode.

Thanks.
 
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  • #2
knockout_artist said:
d2u/d2x + 1/2Lu = 0 where L(x)
something seems missing here?
 
  • #3
ok we have to make characteristic equation out of it so I think

from this
2d2u/d2x + 1/2Lu = 0

following equation comes
2λ2 + Lλ = 0
 

1) What is a fractional term in an ODE?

A fractional term in an ODE refers to a term in the differential equation that contains a fractional exponent. For example, a term like x^(3/2) would be considered a fractional term.

2) How do you solve an ODE with a fractional term?

To solve an ODE with a fractional term, you can use a method called fractional calculus. This involves using fractional derivatives and integrals to manipulate the equation and find a solution.

3) Are there any special techniques for solving ODEs with fractional terms?

Yes, there are several techniques that can be used to solve ODEs with fractional terms. Some common methods include the Laplace transform, the Grunwald-Letnikov method, and the Caputo derivative method.

4) Can an ODE with a fractional term have multiple solutions?

Yes, an ODE with a fractional term can have multiple solutions. This is because fractional calculus introduces more flexibility in the equations and allows for a wider range of possible solutions.

5) What are some real-world applications of ODEs with fractional terms?

ODEs with fractional terms have many applications in various fields of science and engineering. Some examples include modeling non-Newtonian fluids, analyzing electrical circuits, and studying diffusion processes.

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