59.1.687 problem 704

Internal problem ID [9859]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 704
Date solved : Monday, January 27, 2025 at 06:15:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (-\frac {1}{4} x -x^{2}\right ) y^{\prime }-\frac {5 y x}{16}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 55

dsolve(x^2*(1-4*x)*diff(y(x),x$2)+((1-(5/4))*x-(6-4*(5/4))*x^2)*diff(y(x),x)+(5/4)*(1-(5/4))*x*y(x)=0,y(x), singsol=all)
 

\[ y = -\frac {2^{{1}/{4}} \left (c_{1} \sqrt {2}\, \left (x -\frac {\sqrt {1-4 x}}{2}-\frac {1}{2}\right ) \sqrt {1+\sqrt {1-4 x}}-2 c_{2} x^{{5}/{4}}\right )}{\left (1+\sqrt {1-4 x}\right )^{{5}/{4}}} \]

✓ Solution by Mathematica

Time used: 4.189 (sec). Leaf size: 129

DSolve[x^2*(1-4*x)*D[y[x],{x,2}]+((1-(5/4))*x-(6-4*(5/4))*x^2)*D[y[x],x]+(5/4)*(1-(5/4))*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\sqrt [4]{4 x-1} \left (5 c_1 \left (\sqrt {4 x-1}-i\right )^{5/4}+i c_2 \left (\sqrt {4 x-1}+i\right )^{5/4}\right ) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {2}{4 K[1]-1}-\frac {1}{4 K[1]}\right )dK[1]\right )}{5 \sqrt [8]{\sqrt {4 x-1}-i} \sqrt [8]{\sqrt {4 x-1}+i}} \]