78.14.20 problem 6 (e)

Internal problem ID [18356]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 19. The Method of Variation of Parameters. Problems at page 135
Problem number : 6 (e)
Date solved : Tuesday, January 28, 2025 at 11:48:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 30

DSolve[x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==x*Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \left (-(x+1) \operatorname {ExpIntegralEi}(-x)-e^{-x}+c_2 x+c_1\right ) \]