60.9.46 problem 1901

Internal problem ID [11900]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1901
Date solved : Monday, January 27, 2025 at 11:44:17 PM
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 )&=x \left (t \right )+y \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: 0.088 (sec). Leaf size: 47

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

\begin{align*} x \left (t \right ) &= c_{2} +c_3 \,{\mathrm e}^{t} \\ y \left (t \right ) &= c_3 \,{\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{2} \\ z &= c_3 \,{\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_3 \,{\mathrm e}^{t}-c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 93

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

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