59.1.133 problem 135

Internal problem ID [9305]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 135
Date solved : Monday, January 27, 2025 at 06:01:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 26

dsolve(x^2*(1-2*x)*diff(y(x),x$2)-x*(5-4*x)*diff(y(x),x)+(9-4*x)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.212 (sec). Leaf size: 95

DSolve[x^2*(1-2*x)*D[y[x],{x,2}]-x*(5-4*x)*D[y[x],x]+(9-4*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\left (\frac {6}{2 K[2]-1}-\frac {5}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]-4 K[1]^2}dK[1]\right )dK[3]+c_1\right ) \]