23.3.588 problem 596

Internal problem ID [6302]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 596
Date solved : Friday, October 03, 2025 at 02:00:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (4 k^{2}-\left (-p^{2}+1\right ) \sinh \left (x \right )^{2}\right ) y+4 \cosh \left (x \right ) \sinh \left (x \right ) y^{\prime }+4 \sinh \left (x \right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 72

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

\[ y = \sinh \left (x \right )^{k} \left (c_1 \operatorname {hypergeom}\left (\left [-\frac {p}{4}+\frac {1}{4}+\frac {k}{2}, \frac {p}{4}+\frac {1}{4}+\frac {k}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )+c_2 \cosh \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {p}{4}+\frac {k}{2}, \frac {3}{4}+\frac {p}{4}+\frac {k}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.879 (sec). Leaf size: 123

ode=-((4*k^2 - (1 - p^2)*Sinh[x]^2)*y[x]) + 4*Cosh[x]*Sinh[x]*D[y[x],x] + 4*Sinh[x]^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (2 k+p+1),\frac {1}{4} (2 k+p+3),k+1,\tanh ^2(x)\right )+c_2 \operatorname {Hypergeometric2F1}\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)}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
p = symbols("p") 
y = Function("y") 
ode = Eq((-4*k**2 + (1 - p**2)*sinh(x)**2)*y(x) + 4*sinh(x)**2*Derivative(y(x), (x, 2)) + 4*sinh(x)*cosh(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(k**2*y(x) - (-p**2*y(x) + y(x) + 4*Derivative(y(x), (x, 2)))*s