63.13.1 problem 1

Internal problem ID [13172]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page 125
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 05:11:58 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} x&={\mathrm e}^{-\frac {t^{2}}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 22

dsolve([diff(x(t),t$2)+t*diff(x(t),t)+x(t)=0,exp(-t^2/2)],singsol=all)
 

\[ x \left (t \right ) = \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 41

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

\[ x(t)\to \frac {1}{2} e^{-\frac {t^2}{2}} \left (\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {t}{\sqrt {2}}\right )+2 c_2\right ) \]