47.5.1 problem 1

Internal problem ID [7490]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 1
Date solved : Monday, January 27, 2025 at 03:02:09 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+y&=2 x \,{\mathrm e}^{x}-1 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 53

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

\[ y = 2 i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {x^{2}}{2}-\frac {1}{2}} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +1\right )}{2}\right )+\left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+c_{2} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}+2 \,{\mathrm e}^{x}-1 \]

✓ Solution by Mathematica

Time used: 60.111 (sec). Leaf size: 53

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

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