4.36.20 $-y(x)(4k^2-(1-p^2){\sinh }^2(x))+4{\sinh }^2(x)y^{″}(x)+4\sinh (x)\cosh (x)y^{\prime }(x)=0$

ODE
$$-y(x)(4k^2-(1-p^2){\sinh }^2(x))+4{\sinh }^2(x)y^{″}(x)+4\sinh (x)\cosh (x)y^{\prime }(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 1.51321 (sec), leaf count = 123

$$\left\{\{y(x)\to \frac{(-1{)}^{-k}\tanh (x){\tanh }^2(x{)}^{\frac{1}{2}(-k-1)}{(-{\text{sech}}^2(x))}^{\frac{p+2}{4}}(c_1(-1{)}^k{\tanh }^2(x{)}^k{}_2{F}_1(\frac{1}{4}(2k+p+1),\frac{1}{4}(2k+p+3);k+1;{\tanh }^2(x))+c_2{}_2{F}_1(\frac{1}{4}(-2k+p+1),\frac{1}{4}(-2k+p+3);1-k;{\tanh }^2(x)))}{\sqrt[4]{{\text{sech}}^2(x)}}\}\right\}$$

Maple ✓
cpu = 0.294 (sec), leaf count = 92

$$\{y\left(x\right)={(\sinh \left(x\right))}^k(\sinh \left(2x\right){}_2\text{F}{}_1(\frac{3}{4}-\frac{p}{4}+\frac{k}{2},\frac{3}{4}+\frac{p}{4}+\frac{k}{2};\frac{3}{2};\frac{\cosh \left(2x\right)}{2}+\frac{1}{2})\mathit{\_}\mathit{C}\mathit{1}+{}_2\text{F}{}_1(-\frac{p}{4}+\frac{1}{4}+\frac{k}{2},\frac{p}{4}+\frac{1}{4}+\frac{k}{2};\frac{1}{2};\frac{\cosh \left(2x\right)}{2}+\frac{1}{2})\sqrt{-2+2\cosh \left(2x\right)}\mathit{\_}\mathit{C}\mathit{2})\frac{1}{\sqrt{-1+\cosh \left(2x\right)}}\}$$ 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),'implicit')

Maple raw output

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