Internal problem ID [1715]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+t y^{\prime }+y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 27
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)
\[ y \relax (t ) = -\frac {i {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 19
DSolve[{y''[t]+t*y'[t]+y[t]==0,{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \sqrt {2} F\left (\frac {t}{\sqrt {2}}\right ) \\ \end{align*}