59.1.788 problem 810

Internal problem ID [9960]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 810
Date solved : Monday, January 27, 2025 at 06:16:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 22

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

\[ u = {\mathrm e}^{\frac {x^{2}}{2}} c_{1} +c_{2} {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 24

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

\[ u(x)\to e^{\frac {x^2}{2}} \left (c_2 e^x+c_1\right ) \]