60.10.23 problem 1936

Internal problem ID [11934]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1936
Date solved : Tuesday, January 28, 2025 at 06:24:10 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right )\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right ) \left (z \left (t \right )^{2}+x \left (t \right )^{2}\right )\\ \frac {d}{d t}z \left (t \right )&=z \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 1.324 (sec). Leaf size: 700

dsolve([diff(x(t),t)=x(t)*(y(t)^2-z(t)^2),diff(y(t),t)=-y(t)*(z(t)^2+x(t)^2),diff(z(t),t)=z(t)*(x(t)^2+y(t)^2)],singsol=all)
 

\begin{align*} \\ \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}, y \left (t \right ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \left \{z = \frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}, z = -\frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}\right \}\right ] \\ \\ \left [\left \{x \left (t \right ) &= \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}, x \left (t \right ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}\right \}, \{y \left (t \right ) = 0\}, \left \{z = \frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}, z = -\frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_{2} \textit {\_a}^{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} +t +c_3 \right ), x \left (t \right ) = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_{2} \textit {\_a}^{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} +t +c_3 \right )\right \}, \left \{y \left (t \right ) = -\frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )-\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {-2 x \left (t \right ) \left (x \left (t \right )^{3}-\frac {d}{d t}x \left (t \right )+\sqrt {x \left (t \right )^{6}-\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) x \left (t \right )+2 \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}\right \}, \left \{z = \frac {\sqrt {x \left (t \right ) \left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}, z = -\frac {\sqrt {x \left (t \right ) \left (x \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )\right )}}{x \left (t \right )}\right \}\right ] \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved