Internal problem ID [651]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation ,
page 164
Problem number: 44.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+t y^{\prime }+{\mathrm e}^{-t^{2}} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 65
dsolve(diff(y(t),t$2)+ t*diff(y(t),t)+exp(-t^2)*y(t) = 0,y(t), singsol=all)
\[ y \relax (t ) = c_{1} \sin \left (\frac {\sqrt {2}\, {\mathrm e}^{\frac {t^{2}}{2}} \sqrt {\pi }\, \erf \left (\frac {t \sqrt {2}}{2}\right )}{2 \sqrt {{\mathrm e}^{t^{2}}}}\right )+c_{2} \cos \left (\frac {\sqrt {2}\, {\mathrm e}^{\frac {t^{2}}{2}} \sqrt {\pi }\, \erf \left (\frac {t \sqrt {2}}{2}\right )}{2 \sqrt {{\mathrm e}^{t^{2}}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 102
DSolve[y''[t]+t*y'[t]+exp(-t^2)*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to e^{-\frac {1}{4} \left (\sqrt {4 \exp +1}+1\right ) t^2} \left (c_1 \text {HermiteH}\left (-\frac {1}{2 \sqrt {4 \exp +1}}-\frac {1}{2},\frac {\sqrt [4]{4 \exp +1} t}{\sqrt {2}}\right )+c_2 \, _1F_1\left (\frac {1}{4} \left (1+\frac {1}{\sqrt {4 \exp +1}}\right );\frac {1}{2};\frac {1}{2} \sqrt {4 \exp +1} t^2\right )\right ) \\ \end{align*}