54.3.269 problem 1285

Internal problem ID [12564]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1285
Date solved : Friday, October 03, 2025 at 03:38:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 56

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

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

✓ Mathematica. Time used: 13.077 (sec). Leaf size: 210

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

\begin{align*} y(x)&\to \sqrt [4]{1-4 x} \left (-\sqrt {4 x-1}+i\right )^{\frac {1}{2}-\frac {1}{2} i \sqrt {-(a-1)^2}} \left (\sqrt {4 x-1}+i\right )^{\frac {1}{2}+\frac {1}{2} i \sqrt {-(a-1)^2}} \left (c_2 \int _1^x-\frac {\left (i-\sqrt {4 K[2]-1}\right )^{i \sqrt {-(a-1)^2}} \left (\sqrt {4 K[2]-1}+i\right )^{-i \sqrt {-(a-1)^2}}}{4 \sqrt {1-4 K[2]} K[2]}dK[2]+c_1\right ) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {a}{K[1]}+\frac {2}{4 K[1]-1}\right )dK[1]\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None