18.2.9 problem Problem 15.9(b)

Internal problem ID [3492]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.9(b)
Date solved : Sunday, March 30, 2025 at 01:44:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} -\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}&=2 a^{2} \end{align*}

✓ Maple. Time used: 0.258 (sec). Leaf size: 56

ode:=-1/y(x)^2*diff(y(x),x)^2+1/y(x)*diff(diff(y(x),x),x)+2*a*coth(2*a*x)/y(x)*diff(y(x),x) = 2*a^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-a x -\frac {c_2}{a}} \sqrt {{\mathrm e}^{4 a x}-1}\, \left (1-{\mathrm e}^{2 a x}\right )^{-\frac {c_1}{2 a}} \left ({\mathrm e}^{2 a x}+1\right )^{\frac {c_1}{2 a}} \]

✓ Mathematica. Time used: 60.489 (sec). Leaf size: 287

ode=D[1/y[x]*D[y[x],x],x]+(2*a*Coth[1/y[x]*D[y[x],x]])==2*a^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \exp \left (\frac {-\operatorname {PolyLog}\left (2,\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1] \log \left (1-\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+(a+1) \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]{}^2}{4 a \left (a^2-1\right )}\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE -(a*y(x) + sqrt((-2*a**2*y(x)*tanh(2*a*x)**2 + a**2*y(x) + tanh(2*a*x)**2*Derivative(y(x), (x, 2)))*y(x)))/tanh(2*a*x) + Derivative(y(x), x) cannot be solved by the factorable group method