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I am trying to compute the eigenvalues and eigenfunctions of the operator $$L(y)=y''(x)+y'(x), \ \ \ \ 0<x<\pi$$ $$y(0)=0$$ $$y(\pi )=0$$ using the DEigensystem command and I am faced with two issues.

First I can't add the boundary values in DEigensystem, Mahtematica doesn't compute anything from the follwing code

eigen = DEigensystem[{Y''[x] + Y'[x], Y[0] == 0, Y[π] == 0},Y[x], {x, 0, π}, 6]

and if I remove the boundary values, I can't Plot the eigenfnctions, the following code doesn't work

eigen = DEigensystem[Y''[x] + Y'[x],Y[x], {x, 0, π}, 6]

Plot[Evaluate[eigen[[2]][[1]]], {x, 0, π}]

Am I missing something? How can I plot the eigenfunctions of $L$ using DEigensystem?

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1 Answer 1

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To solve your system using DEigensystem, you should specify the boundary conditions with DirichletCondition

L[y_][x_] := Derivative[2][y][x] + Derivative[1][y][x]
{eigenvalues, eigenfunctions} = DEigensystem[{
    L[y][x],
    DirichletCondition[y[x] == 0, x == 0],
    DirichletCondition[y[x] == 0, x == \[Pi]]
    }, y[x], {x, 0, \[Pi]}, 4];

The result is

eigenvalues == {-5/4, -17/4, -37/4, -65/4, -101/4, -145/4}
eigenfunctions == {Sin[x]/E^(x/2), (Cos[x]*Sin[x])/E^(x/2), 
    Sin[3*x]/E^(x/2), Sin[4*x]/E^(x/2), 
    Sin[5*x]/E^(x/2), Sin[6*x]/E^(x/2)}

Then you can Plot[eigenfunctions, {x, 0, \[Pi]}] to get eigenfunctions

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