Solve a Basic Sturm–Liouville Problem

Solve an eigenvalue problem with Dirichlet conditions.

In[1]:=

sol = DSolve[{y''[x] + \[Lambda] y[x] == 0, y[0] == 0, y[\[Pi]] == 0}, y[x], x]

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Make a table of the first 5 eigenfunctions.

In[2]:=

eigfuns = Table[y[x] /. sol[[1]] //. {\[FormalN] -> i, \[Lambda] -> \[FormalN]^2} /. {C[ 1] -> 1}, {i, 5}]

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Plot the eigenfunctions.

In[3]:=

Plot[Evaluate[eigfuns], {x, 0, Pi}]

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Solve an eigenvalue problem with Neumann conditions.

In[4]:=

sol = DSolve[{y''[x] + \[Lambda] y[x] == 0, y'[0] == 0, y'[\[Pi]] == 0}, y[x], x]

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Make a table of the first 5 eigenfunctions.

In[5]:=

eigfuns = Table[y[x] /. sol[[1]] //. {\[FormalN] -> i, \[Lambda] -> \[FormalN]^2} /. {C[ 1] -> 1}, {i, 5}]

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Plot the eigenfunctions.

In[6]:=

Plot[Evaluate[eigfuns], {x, 0, Pi}]

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