59.1.46 problem 48

Internal problem ID [9218]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 48
Date solved : Monday, January 27, 2025 at 05:52:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 49

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

\[ y = \frac {3 x c_{2} \left (x^{2}+6 x +3\right ) \ln \left (x \right )+c_{1} x^{3}+\left (6 c_{1} +51 c_{2} \right ) x^{2}+\left (3 c_{1} +48 c_{2} \right ) x +c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.853 (sec). Leaf size: 119

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

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