54.3.319 problem 1336

Internal problem ID [12614]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1336
Date solved : Wednesday, October 01, 2025 at 02:15:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-3 x +1\right ) y}{\left (x -1\right ) \left (2 x -1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 42

ode:=diff(diff(y(x),x),x) = -(-3*x+1)/(x-1)/(2*x-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (-2 c_2 \left (x -1\right ) \ln \left (2 x -1\right )+2 c_2 \left (x -1\right ) \ln \left (x -1\right )+c_1 x -c_1 +c_2 \right ) \sqrt {2 x -1} \]

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 70

ode=D[y[x],{x,2}] == -(((1 - 3*x)*y[x])/((-1 + x)*(-1 + 2*x)^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.319 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 3*x)*y(x)/((x - 1)*(2*x - 1)**2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} \left (- \log {\left (\frac {2 x - 1}{2 \left (x - 1\right )} \right )} + \frac {2 x - 1}{2 \left (x - 1\right )}\right )\right ) \left (x - 1\right ) \sqrt {x - \frac {1}{2}} {{}_{2}F_{1}\left (\begin {matrix} -1, 0 \\ 1 \end {matrix}\middle | {\frac {2 x - 1}{2 \left (x - 1\right )}} \right )} \]