59.1.135 problem 137

Internal problem ID [9307]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 137
Date solved : Monday, January 27, 2025 at 06:01:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 50

dsolve(2*x^2*(1+x)*diff(y(x),x$2)-x*(6-x)*diff(y(x),x)+(8-x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {2 x^{2} \left (\frac {3 \ln \left (\sqrt {x +1}-1\right ) c_{2}}{2}-\frac {3 \ln \left (\sqrt {x +1}+1\right ) c_{2}}{2}+\left (x +4\right ) c_{2} \sqrt {x +1}+\frac {3 c_{1}}{2}\right )}{3 \left (x +1\right )^{{5}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.244 (sec). Leaf size: 109

DSolve[2*x^2*(1+x)*D[y[x],{x,2}]-x*(6-x)*D[y[x],x]+(8-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {2-K[1]}{4 K[1]^2+4 K[1]}dK[1]-\frac {1}{2} \int _1^x\left (\frac {7}{2 (K[2]+1)}-\frac {3}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2-K[1]}{4 K[1]^2+4 K[1]}dK[1]\right )dK[3]+c_1\right ) \]