59.1.262 problem 265

Internal problem ID [9434]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 265
Date solved : Monday, January 27, 2025 at 06:02:47 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 y x}{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 46

dsolve(x^2*(1-4*x)*diff(y(x),x$2)+((1-(3/2))*x-(6-4*(3/2))*x^2)*diff(y(x),x)+(3/2)*(1-(3/2))*x*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[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,y[x],x,IncludeSingularSolutions -> True]
 

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