13.6.2 problem 6(d)

Internal problem ID [2363]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.1, second order linear differential equations. Page 134
Problem number : 6(d)
Date solved : Tuesday, March 04, 2025 at 02:08:14 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+t y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+t*diff(y(t),t)+y(t) = 0; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {i {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 32

ode=D[y[t],{t,2}]+t*D[y[t],t]+y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sqrt {\frac {\pi }{2}} e^{-\frac {t^2}{2}} \text {erfi}\left (\frac {t}{\sqrt {2}}\right ) \]

✓ Sympy. Time used: 0.805 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t \left (1 - \frac {t^{2}}{3}\right ) + O\left (t^{6}\right ) \]