23.3.170 problem 172

Internal problem ID [5884]
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 : 172
Date solved : Friday, October 03, 2025 at 01:45:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.088 (sec). Leaf size: 54

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

\[ y = \cosh \left (x \right )^{-\frac {a}{2}} \left (c_1 \operatorname {LegendreP}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 b}}{2}, \tanh \left (x \right )\right )+c_2 \operatorname {LegendreQ}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 b}}{2}, \tanh \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.293 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to \left (-\text {sech}^2(x)\right )^{a/4} \left (c_1 P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 b}}(\tanh (x))+c_2 Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 b}}(\tanh (x))\right ) \end{align*}

✗ Sympy

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

False