60.3.105 problem 1109

Internal problem ID [11115]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1109
Date solved : Monday, January 27, 2025 at 10:46:07 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 31

dsolve(x*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x)-x*(x+1)*exp(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.346 (sec). Leaf size: 85

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

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