59.1.379 problem 388

Internal problem ID [9551]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 388
Date solved : Monday, January 27, 2025 at 06:04:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 0.197 (sec). Leaf size: 42

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

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