4.36.20 \(-y(x) \left (4 k^2-\left (1-p^2\right ) \sinh ^2(x)\right )+4 \sinh ^2(x) y''(x)+4 \sinh (x) \cosh (x) y'(x)=0\)

ODE
\[ -y(x) \left (4 k^2-\left (1-p^2\right ) \sinh ^2(x)\right )+4 \sinh ^2(x) y''(x)+4 \sinh (x) \cosh (x) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 1.48737 (sec), leaf count = 123

\[\left \{\left \{y(x)\to \frac {(-1)^{-k} \tanh (x) \tanh ^2(x)^{\frac {1}{2} (-k-1)} \left (-\text {sech}^2(x)\right )^{\frac {p+2}{4}} \left (c_1 (-1)^k \tanh ^2(x)^k \, _2F_1\left (\frac {1}{4} (2 k+p+1),\frac {1}{4} (2 k+p+3);k+1;\tanh ^2(x)\right )+c_2 \, _2F_1\left (\frac {1}{4} (-2 k+p+1),\frac {1}{4} (-2 k+p+3);1-k;\tanh ^2(x)\right )\right )}{\sqrt [4]{\text {sech}^2(x)}}\right \}\right \}\]

Maple ✓
cpu = 1.249 (sec), leaf count = 81

\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-\frac {p}{4}+\frac {1}{4}+\frac {k}{2}, \frac {p}{4}+\frac {1}{4}+\frac {k}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\sinh ^{k}\left (x \right )\right )+\textit {\_C2} \sqrt {\cosh \left (2 x \right )+1}\, \hypergeom \left (\left [\frac {3}{4}-\frac {p}{4}+\frac {k}{2}, \frac {3}{4}+\frac {p}{4}+\frac {k}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\sinh ^{k}\left (x \right )\right )\right ]\] Mathematica raw input

DSolve[-((4*k^2 - (1 - p^2)*Sinh[x]^2)*y[x]) + 4*Cosh[x]*Sinh[x]*y'[x] + 4*Sinh[x]^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((-Sech[x]^2)^((2 + p)/4)*Tanh[x]*(Tanh[x]^2)^((-1 - k)/2)*(C[2]*Hyper
geometric2F1[(1 - 2*k + p)/4, (3 - 2*k + p)/4, 1 - k, Tanh[x]^2] + (-1)^k*C[1]*H
ypergeometric2F1[(1 + 2*k + p)/4, (3 + 2*k + p)/4, 1 + k, Tanh[x]^2]*(Tanh[x]^2)
^k))/((-1)^k*(Sech[x]^2)^(1/4))}}

Maple raw input

dsolve(4*diff(diff(y(x),x),x)*sinh(x)^2+4*diff(y(x),x)*cosh(x)*sinh(x)-(4*k^2-(-p^2+1)*sinh(x)^2)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*hypergeom([-1/4*p+1/4+1/2*k, 1/4*p+1/4+1/2*k],[1/2],1/2*cosh(2*x)+1/
2)*sinh(x)^k+_C2*(cosh(2*x)+1)^(1/2)*hypergeom([3/4-1/4*p+1/2*k, 3/4+1/4*p+1/2*k
],[3/2],1/2*cosh(2*x)+1/2)*sinh(x)^k]