60.9.47 problem 1902

Internal problem ID [11901]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1902
Date solved : Monday, January 27, 2025 at 11:44:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-y \left (t \right )+z \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=t\\ \frac {d}{d t}z \left (t \right )-x \left (t \right )-z \left (t \right )&=t \end{align*}

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 55

dsolve([diff(x(t),t)-y(t)+z(t)=0,diff(y(t),t)-x(t)-y(t)=t,diff(z(t),t)-x(t)-z(t)=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} -t -1 \\ z &= c_3 \,{\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_3 \,{\mathrm e}^{t}-c_{2} -t -1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 246

DSolve[{D[x[t],t]-y[t]+z[t]==0,D[y[t],t]-x[t]-y[t]==t,D[z[t],t]-x[t]-z[t]==t},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \left (e^t-1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+c_2 \left (e^t-1\right )-c_3 \left (e^t-1\right )+c_1 \\ y(t)\to \left (e^t t+1\right ) \int _1^te^{-K[1]} K[1]dK[1]+\left (-e^t (t-1)-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+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 \left (e^t (t-1)+1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t (t-2)+1\right ) \int _1^te^{-K[2]} K[2]dK[2]+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*}