59.1.689 problem 706

Internal problem ID [9861]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 706
Date solved : Monday, January 27, 2025 at 06:15:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 48

dsolve(x^2*diff(y(x),x$2)+x*(x+1)*diff(y(x),x)+(3*x-1)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{2} c_{2} {\mathrm e}^{-x} \left (x -3\right ) \operatorname {Ei}_{1}\left (-x \right )+x^{2} c_{1} \left (x -3\right ) {\mathrm e}^{-x}+c_{2} \left (x^{2}-2 x -1\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.218 (sec). Leaf size: 43

DSolve[x^2*D[y[x],{x,2}]+x*(x+1)*D[y[x],x]+(3*x-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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