Symbolic & Numeric Calculus

Solve a Fredholm Integral Equation

Solve a Fredholm integral equation using DSolveValue.

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eqn = y[x] == Sin[7 x] - x/a + \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), FractionBox[\(\[Pi]\), \(2\)]]\(\(x\ y[t]\) \[DifferentialD]t\)\);
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sol = DSolveValue[eqn, y[x], x]
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Plot the solution.

In[3]:=
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Plot[Table[sol, {a, -1, 4, 0.7}] // Evaluate, {x, 0, Pi/2}]
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Solve a homogeneous Fredholm equation of the second kind.

In[4]:=
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eqn = y[x] == \[Lambda] \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]\)]\(Cos[x + t] y[ t] \[DifferentialD]t\)\);
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sol = DSolveValue[eqn, y[x], x]
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Plot the solution.

In[6]:=
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Plot[Table[ sol /. {C[1] -> 1, C[2] -> 1}, {\[Lambda], {-2/Pi, 2/Pi}}] // Evaluate, {x, 0, 3 \[Pi]}]
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