75.27.6 problem 781

Internal problem ID [17242]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 781
Date solved : Tuesday, January 28, 2025 at 09:58:23 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 55

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

\begin{align*} x \left (t \right ) &= -c_{2} \cos \left (t \right )+\sin \left (t \right ) c_{3} +c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{3} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} \cos \left (t \right )+\sin \left (t \right ) c_{3} \\ z &= c_{1} {\mathrm e}^{t}+c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 112

DSolve[{D[x[t],t]==-y[t]+z[t],D[y[t],t]==z[t],D[z[t],t]==-x[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 \cos (t)+(c_3-c_2) \sin (t) \\ y(t)\to \frac {1}{2} \left ((-c_1+c_2+c_3) e^t+(c_1+c_2-c_3) \cos (t)+(c_1-c_2+c_3) \sin (t)\right ) \\ z(t)\to \frac {1}{2} \left ((-c_1+c_2+c_3) e^t+(c_1-c_2+c_3) \cos (t)-(c_1+c_2-c_3) \sin (t)\right ) \\ \end{align*}