Internal problem ID [7187]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 465.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x +1\right )^{2} y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 19
dsolve((x+1)^2*diff(y(x),x$2)-2*(x+1)*diff(y(x),x)-(x^2+2*x-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sinh \relax (x ) \left (x +1\right )+c_{2} \cosh \relax (x ) \left (x +1\right ) \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 147
DSolve[(x+1)^2*y''[x]-2*(x+1)*x*y'[x]-(x^2+2*x-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{\frac {1}{2} i \left (\sqrt {7}+i\right )} e^{-\left (\left (\sqrt {2}-1\right ) (x+1)\right )} (x+1)^{\frac {1}{2} i \left (\sqrt {7}+i\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (1-\sqrt {2}+i \sqrt {7}\right ),1+i \sqrt {7},2 \sqrt {2} (x+1)\right )+c_2 L_{\frac {1}{2} \left (-1+\sqrt {2}-i \sqrt {7}\right )}^{i \sqrt {7}}\left (2 \sqrt {2} (x+1)\right )\right ) \\ \end{align*}