59.1.597 problem 613

Internal problem ID [9769]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 613
Date solved : Monday, January 27, 2025 at 06:14:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 82

dsolve(x^2*(1+x)*diff(y(x),x$2)+x*(1-10*x)*diff(y(x),x)-(9-10*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {8 c_{2} x^{13}+91 c_{2} x^{12}+468 c_{2} x^{11}+1430 c_{2} x^{10}+2860 c_{2} x^{9}+3861 c_{2} x^{8}+3432 c_{2} x^{7}+1716 c_{2} x^{6}+715 c_{1} x^{4}+572 c_{1} x^{3}+234 c_{1} x^{2}+52 c_{1} x +5 c_{1}}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.472 (sec). Leaf size: 123

DSolve[x^2*(1+x)*D[y[x],{x,2}]+x*(1-10*x)*D[y[x],x]-(9-10*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{8} (8 x-5) \exp \left (\int _1^x\frac {8 K[1]-5}{2 K[1] (K[1]+1)}dK[1]-\frac {1}{2} \int _1^x\left (\frac {1}{K[2]}-\frac {11}{K[2]+1}\right )dK[2]\right ) \left (c_2 \int _1^x\frac {64 \exp \left (-2 \int _1^{K[3]}\frac {8 K[1]-5}{2 K[1] (K[1]+1)}dK[1]\right )}{(5-8 K[3])^2}dK[3]+c_1\right ) \]