59.1.447 problem 461

Internal problem ID [9619]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 461
Date solved : Monday, January 27, 2025 at 06:04:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x +1\right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 y \left (1+x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 17

dsolve((2*x+1)*x*diff(y(x),x$2)-2*(2*x^2-1)*diff(y(x),x)-4*(x+1)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} {\mathrm e}^{2 x} x +c_{1}}{x} \]

✓ Solution by Mathematica

Time used: 0.242 (sec). Leaf size: 92

DSolve[(2*x+1)*x*D[y[x],{x,2}]-2*(2*x^2-1)*D[y[x],x]-4*(x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]-\frac {1}{2} \int _1^x\frac {2-4 K[2]^2}{2 K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]\right )dK[3]+c_1\right ) \]