67.2.54 problem Problem 20(a)

Internal problem ID [13940]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 20(a)
Date solved : Monday, March 31, 2025 at 08:19:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.151 (sec). Leaf size: 80

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

\[ y = -\frac {\left (\left (\frac {2 \left (-1+\operatorname {erf}\left (\frac {\sqrt {4 x -2}}{2}\right )\right ) c_2}{3}+\operatorname {erf}\left (\frac {\sqrt {4 x -2}}{2}\right ) c_1 \right ) \sqrt {\pi }\, \sqrt {2}\, \left (x -\frac {1}{2}\right ) {\mathrm e}^{-\frac {1}{4}} \sqrt {2 x -1}+2 \,{\mathrm e}^{\frac {1}{4}-x} \left (c_1 +\frac {2 c_2}{3}\right ) \left (x -1\right )\right ) 2^{{1}/{4}}}{\sqrt {2 x -1}} \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 88

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]+2}{2 K[1]-1}dK[1]-\frac {1}{2} \int _1^x\left (1+\frac {1}{2 K[2]-1}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]+2}{2 K[1]-1}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

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

False