23.3.55 problem 57

Internal problem ID [5769]
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 : 57
Date solved : Friday, October 03, 2025 at 01:43:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a +b \sinh \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 39

ode:=(a+b*sinh(x)^2)*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {MathieuC}\left (\frac {b}{2}-a , \frac {b}{4}, i x \right )+c_2 \operatorname {MathieuS}\left (\frac {b}{2}-a , \frac {b}{4}, i x \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to c_1 \text {MathieuC}\left [\frac {1}{2} (b-2 a),\frac {b}{4},i x\right ]-c_2 \text {MathieuS}\left [\frac {1}{2} (b-2 a),\frac {b}{4},i x\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((a + b*sinh(x)**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False