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Solve this third-order differential equations using DSolve and Integrate?

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Hi,

I solve this third-order differential equations and obtain f1(y)

Clear["Global`*"]
f0[y_] := ((Cosh[a0 t] - Cosh[y t]) Csch[a0 t])/t
DSolve[{f1'''[y] - f1'[y] Cos[\[Alpha]] + (3 f0'[y])/(2 f0[y]) == 0, 
  f1'[0] == 0, f1[a0] + a1 f0'[a0] == 0, f1'[a0] + a1 f0''[a0] == 0}, 
 f1[y], y]

By substituting f0(y) and f1(y) into this formula

int1 = Integrate[(f0[y])^2 (f1[y]), {y, 0, a0}]

and

Solve[int==0,a1]

I want to find a1 but I did not get it. Did I miss something? Please help me. Thank you

POSTED BY: Nurul Ainina
Answer
22 days ago

Using the result from DSolve is not intuitive for beginners. Try this:

f0[y_] := ((Cosh[a0 t] - Cosh[y t]) Csch[a0 t])/t;
sol = DSolve[{f1'''[y] - 
      f1'[y] Cos[\[Alpha]] + (3 f0'[y])/(2 f0[y]) == 0, f1'[0] == 0, 
    f1[a0] + a1 f0'[a0] == 0, f1'[a0] + a1 f0''[a0] == 0}, f1[y], y];
int1 = Integrate[(f0[y])^2 (f1[y] /. sol[[1]]), {y, 0, a0}]

or look up the documentation for DSolveValue.

POSTED BY: Gianluca Gorni
Answer
22 days ago

Thank you for your suggestion. I tried to run but again it took time and I guess Mathematica would not give answer.

POSTED BY: Nurul Ainina
Answer
21 days ago

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