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