67.16.19 problem 24.4 (a)

Internal problem ID [16818]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.4 (a)
Date solved : Thursday, October 02, 2025 at 01:39:22 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y&={\mathrm e}^{-x^{2}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 53

ode:=x^3*diff(diff(diff(y(x),x),x),x)-3*x^2*diff(diff(y(x),x),x)+6*x*diff(y(x),x)-6*y(x) = exp(-x^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x^{2}} \left (2 x^{2}-1\right )}{6}+x \left (-\frac {x \,\operatorname {Ei}_{1}\left (x^{2}\right )}{2}+\operatorname {erf}\left (x \right ) \left (\frac {x^{2}}{3}-\frac {1}{2}\right ) \sqrt {\pi }+c_3 \,x^{2}+c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 77

ode=x^3*D[y[x],{x,3}]-3*x^2*D[y[x],{x,2}]+6*x*D[y[x],x]-6*y[x]==Exp[-x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \left (\sqrt {\pi } \left (2 x^2-3\right ) x \text {erf}(x)+3 x^2 \operatorname {ExpIntegralEi}\left (-x^2\right )+6 c_3 x^3+2 e^{-x^2} x^2-e^{-x^2}+6 c_2 x^2+6 c_1 x\right ) \end{align*}

✓ Sympy. Time used: 1.228 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) - 3*x**2*Derivative(y(x), (x, 2)) + 6*x*Derivative(y(x), x) - 6*y(x) - exp(-x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \frac {\sqrt {\pi } x^{3} \operatorname {erf}{\left (x \right )}}{3} + \frac {x^{2} \operatorname {Ei}{\left (x^{2} e^{i \pi } \right )}}{2} + \frac {x^{2} e^{- x^{2}}}{3} - \frac {\sqrt {\pi } x \operatorname {erf}{\left (x \right )}}{2} - \frac {e^{- x^{2}}}{6} \]