61.2.42 problem Problem 18(g)

Internal problem ID [15280]
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 : Thursday, October 02, 2025 at 10:09:22 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 83

ode:=diff(diff(y(x),x),x)+x^2*diff(y(x),x)+2*x*y(x) = 2*x; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 59

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

\begin{align*} 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 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 2*x*y(x) - 2*x + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x*(1 - y(x)) - Derivative(y(x), (x, 2)))/x**2 cannot be solved by the factorable group method