a=1; eqns[a?NumericQ, r?NumericQ, gamma_?NumericQ] := {Derivative[4][f][y] - 2 Derivative[2][f][y] a^2 + a^2 (-r y + a^2) f[y] == 0, f[0] == 0, Derivative[3][f][0] - Derivative[1][f][0] a^2 == 0,
Derivative[1][f][0] == 1, Derivative[2][f][0] == gamma}

    sol[a_?NumericQ, r_?NumericQ, \[Gamma]_?NumericQ] :=   NDSolve[eqns[a,
    r, \[Gamma]], f, {y, 0, 1}, Method -> "BDF",    MaxSteps -> Infinity,
    InterpolationOrder -> All]

    eval = sol[1, 2, 3][[1, 1]]

    ff = f /. eval

    Plot[ff[y], {y, 0, 1}]

Hi everyone, I would know how I can find the value of 'r' using FindRoot and plot it (r,a) with this conditions target y=1,f[1] = 0, f''[1] = 0. Thanks a lot.