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

Posted 2 years ago
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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
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Posted 2 years 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`.