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 : Saturday, March 29, 2025 at 10:51:40 PM
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*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 65

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

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

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

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

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

✓ Sympy. Time used: 1.096 (sec). Leaf size: 61

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4} e^{- t^{2}}}{12} + \frac {t^{4} e^{- 2 t^{2}}}{24} - \frac {t^{2} e^{- t^{2}}}{2} + 1\right ) + C_{1} t \left (- \frac {t^{2}}{6} - \frac {t^{2} e^{- t^{2}}}{6} + 1\right ) + O\left (t^{6}\right ) \]