Internal problem ID [7570]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {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}}=0} \end {gather*}
✓ Solution by Maple
Time used: 2.032 (sec). Leaf size: 90
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)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right ) \left (\int \frac {x^{3} {\mathrm e}^{\frac {-x^{3}+2 x^{2}+2}{x}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.245 (sec). Leaf size: 79
DSolve[y''[x]== (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]
\begin{align*} 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}} \\ \end{align*}