57.4.18 problem 8

Internal problem ID [14342]
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 1.3.1 Separable equations. Exercises page 26
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:31:56 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} x \left (0\right )&=\ln \left (2\right ) \\ \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 15

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

\[ x = -\ln \left (3\right )+\ln \left (t^{3}+6\right ) \]

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 15

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

\begin{align*} x(t)&\to \log \left (\frac {1}{3} \left (t^3+6\right )\right ) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 10

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

\[ x{\left (t \right )} = \log {\left (\frac {t^{3}}{3} + 2 \right )} \]