Internal problem ID [13475]
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).
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 y^{\prime } x -6 y={\mathrm e}^{-x^{2}}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 73
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)
\[ y = \frac {\left (2 \sqrt {\pi }\, \operatorname {erf}\left (x \right ) {\mathrm e}^{x^{2}} x^{3}-3 \sqrt {\pi }\, \operatorname {erf}\left (x \right ) {\mathrm e}^{x^{2}} x -3 \,\operatorname {expIntegral}_{1}\left (x^{2}\right ) {\mathrm e}^{x^{2}} x^{2}+2 x^{2}-1\right ) {\mathrm e}^{-x^{2}}}{6}+c_{3} x^{3}+c_{2} x^{2}+c_{1} x \]
✓ Solution by Mathematica
Time used: 0.068 (sec). Leaf size: 77
DSolve[x^3*y'''[x]-3*x^2*y''[x]+6*x*y'[x]-6*y[x]==Exp[-x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{6} \left (\sqrt {\pi } \left (2 x^2-3\right ) x \text {erf}(x)+3 x^2 \operatorname {ExpIntegralEi}\left (-x^2\right )+6 c_3 x^3+2 e^{-x^2} x^2-e^{-x^2}+6 c_2 x^2+6 c_1 x\right ) \]