15.8.7 problem 7

Internal problem ID [3010]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 03:43:12 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} 2 \,{\mathrm e}^{x}-t^{2}+t \,{\mathrm e}^{x} x^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 19

ode:=2*exp(x(t))-t^2+t*exp(x(t))*diff(x(t),t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = -2 \ln \left (2\right )+\ln \left (\frac {t^{4}+c_{1}}{t^{2}}\right ) \]

✓ Mathematica. Time used: 1.875 (sec). Leaf size: 20

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

\[ x(t)\to \log \left (\frac {t^2}{4}+\frac {c_1}{t^2}\right ) \]

✓ Sympy. Time used: 0.669 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = \log {\left (\frac {C_{1} + \frac {t^{4}}{4}}{t^{2}} \right )} \]