57.13.1 problem 1

Internal problem ID [14453]
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 : Thursday, October 02, 2025 at 09:37:28 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime } t +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*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 22

ode:=diff(diff(x(t),t),t)+t*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) c_1 +c_2 \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 41

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

\begin{align*} 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 ) \end{align*}

✗ Sympy

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

False