59.1.262 problem 265

Internal problem ID [9434]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 265
Date solved : Sunday, March 30, 2025 at 02:34:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-4 x \right ) y^{\prime \prime }-\frac {x y^{\prime }}{2}-\frac {3 x y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 46

ode:=x^2*(1-4*x)*diff(diff(y(x),x),x)-1/2*x*diff(y(x),x)-3/4*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 4.075 (sec). Leaf size: 125

ode=x^2*(1-4*x)*D[y[x],{x,2}]+((1-(3/2))*x-(6-4*(3/2))*x^2)*D[y[x],x]+(3/2)*(1-(3/2))*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\sqrt [4]{4 x-1} \left (6 c_1 \left (\sqrt {4 x-1}-i\right )^{3/2}+i c_2 \left (\sqrt {4 x-1}+i\right )^{3/2}\right ) \exp \left (-\frac {1}{2} \int _1^x-\frac {1}{2 K[1]-8 K[1]^2}dK[1]\right )}{6 \sqrt [4]{\sqrt {4 x-1}-i} \sqrt [4]{\sqrt {4 x-1}+i}} \]

✓ Sympy. Time used: 0.433 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 4*x)*Derivative(y(x), (x, 2)) - 3*x*y(x)/4 - x*Derivative(y(x), x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (4 x - 1\right )^{\frac {3}{4}} \left (C_{1} \left (\frac {x}{4 x - 1}\right )^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {5}{2} \end {matrix}\middle | {\frac {4 x}{4 x - 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {4 x}{4 x - 1}} \right )}\right ) \]