10.8.29 problem 44

Internal problem ID [1301]
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
Date solved : Monday, January 27, 2025 at 04:50:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+t y^{\prime }+{\mathrm e}^{-t^{2}} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 55

dsolve(diff(y(t),t$2)+ t*diff(y(t),t)+exp(-t^2)*y(t) = 0,y(t), singsol=all)
 

\[ y = c_1 \,\operatorname {csgn}\left ({\mathrm e}^{\frac {t^{2}}{2}}\right ) \sin \left (\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )}{2}\right )+c_2 \cos \left (\frac {\sqrt {2}\, \operatorname {csgn}\left ({\mathrm e}^{\frac {t^{2}}{2}}\right ) \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 102

DSolve[D[y[t],{t,2}]+t*D[y[t],t]+exp(-t^2)*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-\frac {1}{4} \left (\sqrt {4 \exp +1}+1\right ) t^2} \left (c_1 \operatorname {HermiteH}\left (-\frac {1}{2}-\frac {1}{2 \sqrt {4 \exp +1}},\frac {\sqrt [4]{4 \exp +1} t}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (1+\frac {1}{\sqrt {4 \exp +1}}\right ),\frac {1}{2},\frac {1}{2} \sqrt {4 \exp +1} t^2\right )\right ) \]