14.31.3 problem 10

Internal problem ID [2822]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.6 (Qualitative properties of orbits). Page 417
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 02:50:00 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} z^{\prime \prime }+{\mathrm e}^{z^{2}}&=1 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 59

ode:=diff(diff(z(t),t),t)+exp(z(t)^2) = 1; 
dsolve(ode,z(t), singsol=all);
 

\begin{align*} \int _{}^{z}\frac {1}{\sqrt {-\sqrt {\pi }\, \operatorname {erfi}\left (\textit {\_a} \right )+c_1 +2 \textit {\_a}}}d \textit {\_a} -t -c_2 &= 0 \\ -\int _{}^{z}\frac {1}{\sqrt {-\sqrt {\pi }\, \operatorname {erfi}\left (\textit {\_a} \right )+c_1 +2 \textit {\_a}}}d \textit {\_a} -t -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 44

ode=D[z[t],{t,2}]+Exp[z[t]^2]==1; 
ic={}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{z(t)}\frac {1}{\sqrt {c_1+2 \left (K[1]-\frac {1}{2} \sqrt {\pi } \text {erfi}(K[1])\right )}}dK[1]{}^2=(t+c_2){}^2,z(t)\right ] \]

✗ Sympy

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(exp(z(t)**2) + Derivative(z(t), (t, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=z(t),ics=ics)
 

Timed Out