54.3.26 problem 1027

Internal problem ID [12321]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1027
Date solved : Friday, October 03, 2025 at 03:18:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.270 (sec). Leaf size: 69

ode:=diff(diff(y(x),x),x)-(n*(n+1)*k^2*JacobiSN(x,k)^2+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {b}{4 k^{2}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right )+c_2 \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {k^{2}+b +1}{4 k^{2}}, 1+\frac {n}{2}, \frac {1}{2}-\frac {n}{2}, \frac {3}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right ) \operatorname {JacobiSN}\left (x , k\right ) \]

✓ Mathematica. Time used: 0.565 (sec). Leaf size: 209

ode=(b + a*JacobiSN[x, k]^2)*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {k \text {sn}(x|k)^2-1} \left (c_1 \text {HeunG}\left [\frac {1}{k},\frac {k-b}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+3\right ),\frac {\sqrt {k} \sqrt {k-4 a}+2 a+k}{2 \left (\sqrt {k} \sqrt {k-4 a}+k\right )},\frac {1}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]+c_2 \text {sn}(x|k) \text {HeunG}\left [\frac {1}{k},\frac {-b+4 k+1}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+5\right ),\frac {\sqrt {k} \sqrt {k-4 a}+a+k}{\sqrt {k} \sqrt {k-4 a}+k},\frac {3}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
b = symbols("b") 
k = symbols("k") 
n = symbols("n") 
y = Function("y") 
ode = Eq((-b - k**2*n*(n + 1)*JacobiSN(x, k)**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : cannot determine truth value of Relational: _n < x