77.1.148 problem 175 (page 265)

Internal problem ID [18038]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 175 (page 265)
Date solved : Tuesday, January 28, 2025 at 11:23:20 AM
CAS classification : system_of_ODEs

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

Solution by Maple

Time used: 0.115 (sec). Leaf size: 175

dsolve([diff(x(t),t)=y(t),diff(y(t),t)=z(t),diff(z(t),t)=x(t)],singsol=all)
 
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +c_{3} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2} \\ z &= c_{1} {\mathrm e}^{t}-\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}+\frac {c_{3} {\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2} \\ \end{align*}

Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 234

DSolve[{D[x[t],t]==y[t],D[y[t],t]==z[t],D[z[t],t]==x[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 
\begin{align*} x(t)\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}+(2 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}-(c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} (c_1-c_3) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ z(t)\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}-(c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1-c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ \end{align*}