59.3.1 problem Kovacic 1985 paper. page 13. section 3.2, example 1

Internal problem ID [10003]
Book : Collection of Kovacic problems
Section : section 3. Problems from Kovacic related papers
Problem number : Kovacic 1985 paper. page 13. section 3.2, example 1
Date solved : Monday, January 27, 2025 at 06:16:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.732 (sec). Leaf size: 66

dsolve(diff(y(x),x$2)= (4*x^6-8*x^5+12*x^4+4*x^3+7*x^2-20*x+4)/(4*x^4)*y(x),y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right ) \left (c_{2} \left (\int \frac {x^{3} {\mathrm e}^{\frac {-x^{3}+2 x^{2}+2}{x}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right )+c_{1} \right )}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.596 (sec). Leaf size: 79

DSolve[D[y[x],{x,2}]== (4*x^6-8*x^5+12*x^4+4*x^3+7*x^2-20*x+4)/(4*x^4)*y[x],y[x],x,IncludeSingularSolutions -> True]
 

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