4.27.46 \(a \tanh (x) y'(x)+b y(x)+y''(x)=0\)

ODE
\[ a \tanh (x) y'(x)+b y(x)+y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.625962 (sec), leaf count = 75

\[\left \{\left \{y(x)\to \left (-\text {sech}^2(x)\right )^{a/4} \left (c_1 P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 b}}(\tanh (x))+c_2 Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 b}}(\tanh (x))\right )\right \}\right \}\]

Maple ✓
cpu = 0.932 (sec), leaf count = 59

\[\left [y \left (x \right ) = \textit {\_C1} \left (\cosh ^{-\frac {a}{2}}\left (x \right )\right ) \LegendreP \left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 b}}{2}, \tanh \left (x \right )\right )+\textit {\_C2} \left (\cosh ^{-\frac {a}{2}}\left (x \right )\right ) \LegendreQ \left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 b}}{2}, \tanh \left (x \right )\right )\right ]\] Mathematica raw input

DSolve[b*y[x] + a*Tanh[x]*y'[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (C[1]*LegendreP[(-2 + a)/2, Sqrt[a^2 - 4*b]/2, Tanh[x]] + C[2]*Legendr
eQ[(-2 + a)/2, Sqrt[a^2 - 4*b]/2, Tanh[x]])*(-Sech[x]^2)^(a/4)}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)*tanh(x)+b*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*cosh(x)^(-1/2*a)*LegendreP(1/2*a-1,1/2*(a^2-4*b)^(1/2),tanh(x))+_C2*
cosh(x)^(-1/2*a)*LegendreQ(1/2*a-1,1/2*(a^2-4*b)^(1/2),tanh(x))]