57.3.7 problem 6

Internal problem ID [14323]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page 19
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:31:31 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} x \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 17

ode:=diff(x(t),t) = exp(-t)/t^(1/2); 
ic:=[x(1) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \left (\operatorname {erf}\left (\sqrt {t}\right )-\operatorname {erf}\left (1\right )\right ) \sqrt {\pi } \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 22

ode=D[x[t],t]==Exp[-t]/Sqrt[t]; 
ic={x[1]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \sqrt {\pi } \left (\text {erf}\left (\sqrt {t}\right )-\text {erf}(1)\right ) \end{align*}

✓ Sympy. Time used: 0.231 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(Derivative(x(t), t) - exp(-t)/sqrt(t),0) 
ics = {x(1): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \sqrt {\pi } \operatorname {erf}{\left (\sqrt {t} \right )} - \sqrt {\pi } \operatorname {erf}{\left (1 \right )} \]