4.8.29 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 : Tuesday, September 30, 2025 at 04:32:23 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*}

✓ 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: 0.697 (sec). Leaf size: 1305

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]
 

Too large to display

✗ Sympy

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)
 

False