65.3.12 problem 8.8

Internal problem ID [13730]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 8, Separable equations. Exercises page 72
Problem number : 8.8
Date solved : Tuesday, January 28, 2025 at 05:59:57 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=-x \left (k^{2}+x^{2}\right ) \end{align*}

With initial conditions

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

✗ Solution by Maple

dsolve([diff(x(t),t)=-x(t)*(k^2+x(t)^2),x(0) = x__0],x(t), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.234 (sec). Leaf size: 54

DSolve[{D[x[t],t]==-x[t]*(k^2+x[t]^2),{x[0]==x0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \text {InverseFunction}\left [\int _0^{\text {$\#$1}}\frac {1}{K[1] \left (k^2+K[1]^2\right )}dK[1]\&\right ]\left [\int _0^{\text {x0}}\frac {1}{K[1]^3+k^2 K[1]}dK[1]-t\right ] \]