14.31.4 problem 11

Internal problem ID [2823]
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 : 11
Date solved : Monday, January 27, 2025 at 06:18:58 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} z^{\prime \prime }+\frac {z}{1+z^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 55

dsolve(diff(z(t),t$2)+z(t)/(1+z(t)^2)=0,z(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.325 (sec). Leaf size: 36

DSolve[D[z[t],{t,2}]+z[t]/(1+z[t]^2)==0,z[t],t,IncludeSingularSolutions -> True]
 

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