73.16.19 problem 24.4 (a)

Internal problem ID [15531]
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 : Tuesday, January 28, 2025 at 08:01:23 AM
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*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 0.087 (sec). Leaf size: 77

DSolve[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],y[x],x,IncludeSingularSolutions -> True]
 

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