59.1.451 problem 465

Internal problem ID [9623]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 465
Date solved : Monday, January 27, 2025 at 06:04:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 17

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)
 

\[ y = \left (x +1\right ) \left (c_{1} \sinh \left (x \right )+c_{2} \cosh \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 146

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (1-\sqrt {2}+i \sqrt {7}\right ),1+i \sqrt {7},2 \sqrt {2} (x+1)\right )+c_2 L_{\frac {1}{2} \left (-1+\sqrt {2}-i \sqrt {7}\right )}^{i \sqrt {7}}\left (2 \sqrt {2} (x+1)\right )\right ) \exp \left (\int _1^x\frac {-2 \sqrt {2} K[1]+2 K[1]+i \sqrt {7}-2 \sqrt {2}+1}{2 K[1]+2}dK[1]\right ) \]