67.2.42 problem Problem 18(g)

Internal problem ID [13999]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 18(g)
Date solved : Tuesday, January 28, 2025 at 06:11:22 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }+2 y x&=2 x \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 83

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

\[ y = \frac {2 x \sqrt {3}\, \pi c_{1} {\mathrm e}^{-\frac {x^{3}}{3}}-3 x \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) c_{1} \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{3}}+\left (-x^{3}\right )^{{1}/{3}}+c_{2} \left (-x^{3}\right )^{{1}/{3}} {\mathrm e}^{-\frac {x^{3}}{3}}-\left (-x^{3}\right )^{{1}/{3}} {\mathrm e}^{-\frac {x^{3}}{3}}}{\left (-x^{3}\right )^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 59

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

\[ y(x)\to c_2 e^{-\frac {x^3}{3}}+\frac {c_1 e^{-\frac {x^3}{3}} \left (-x^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^3}{3}\right )}{3^{2/3} x^2}+1 \]