Boundary Value Problem + Green's Function

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SUMMARY

The discussion centers on solving a boundary value problem (BVP) defined by the differential equation y'' + 4y = f(x) with boundary conditions y(0) = 0 and y'(1) = 0. The user initially attempted to find the Green's function using the trial solution y(x) = Asin(2x) + Bcos(2x) but encountered issues with the boundary conditions. Ultimately, the user resolved their confusion regarding the derivation of the Green's function, indicating a successful understanding of the problem.

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  • Understanding of boundary value problems (BVPs)
  • Familiarity with Green's functions in differential equations
  • Knowledge of trigonometric functions and their derivatives
  • Basic skills in solving ordinary differential equations (ODEs)
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  • Study the derivation of Green's functions for various types of differential equations
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  • Learn about the method of separation of variables for solving BVPs
  • Investigate numerical methods for approximating solutions to BVPs
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Mathematicians, physicists, and engineers involved in solving boundary value problems and applying Green's functions in their work.

sassie
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Consider the BVP

y''+4y=f(x) (0[tex]\leq[/tex]x[tex]\leq[/tex]1)
y(0)=0 y'(1)=0

Find the Green's function (two-sided) for this problem.

Working: So firstly, I let y(x)=Asin2x+Bcos2x

Then using the boundary conditions,

Asin(2.0)+Bcos(2.0)=0 => B=0

y'(x)=2Acos(2x)-2Asin(2x)
y'(0)=2A=0 => A=0

But is this right? How can I derive a Green's function (two-sided) from this? Please help.
 
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Please ignore. I figured out what I did wrong.
 

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