59.1.552 problem 568

Internal problem ID [9724]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 568
Date solved : Monday, January 27, 2025 at 06:13:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.444 (sec). Leaf size: 111

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

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