59.1.151 problem 153

Internal problem ID [9323]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 153
Date solved : Monday, January 27, 2025 at 06:01:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 47

dsolve(x^2*(1-x)*diff(y(x),x$2)-x*(3-5*x)*diff(y(x),x)+(4-5*x)*y(x)=0,y(x), singsol=all)
 

\[ y = x^{2} \left (c_{1} \left (x -1\right )^{3}+c_{2} \left (-\left (x -1\right )^{3} \ln \left (x -1\right )+\left (x -1\right )^{3} \ln \left (x \right )-x^{2}+\frac {5 x}{2}-\frac {11}{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.255 (sec). Leaf size: 104

DSolve[x^2*(1-x)*D[y[x],{x,2}]-x*(3-5*x)*D[y[x],x]+(4-5*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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