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 : Monday, January 27, 2025 at 05:50:16 AM
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*}

✓ Solution by Maple

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

dsolve([diff(y(t),t$2)+t*diff(y(t),t)+y(t)=0,y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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