67.5.17 problem Problem 3(f)

Internal problem ID [14103]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(f)
Date solved : Tuesday, January 28, 2025 at 06:14:29 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )+x \left (t \right )-z \left (t \right )&=0\\ y^{\prime }-y+x \left (t \right )&=0\\ z^{\prime }\left (t \right )+x \left (t \right )+2 y-3 z \left (t \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 47

dsolve([diff(x(t),t)+x(t)-z(t)=0,diff(y(t),t)-y(t)+x(t)=0,diff(z(t),t)+x(t)+2*y(t)-3*z(t)=0],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} t +c_{3} {\mathrm e}^{3 t} \\ y &= -\frac {c_{3} {\mathrm e}^{3 t}}{2}+c_{2} +c_{1} +c_{2} t \\ z \left (t \right ) &= c_{2} +4 c_{3} {\mathrm e}^{3 t}+c_{1} +c_{2} t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 132

DSolve[{D[x[t],t]+x[t]-z[t]==0,D[y[t],t]-y[t]+x[t]==0,D[z[t],t]+x[t]+2*y[t]-3*z[t]==0},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{9} \left (-9 c_1 (t-1)+c_2 \left (6 t-2 e^{3 t}+2\right )+c_3 \left (3 t+2 e^{3 t}-2\right )\right ) \\ y(t)\to \frac {1}{9} \left (-9 c_1 t+c_2 \left (6 t+e^{3 t}+8\right )+c_3 \left (3 t-e^{3 t}+1\right )\right ) \\ z(t)\to \frac {1}{9} \left (-9 c_1 t-2 c_2 \left (-3 t+4 e^{3 t}-4\right )+c_3 \left (3 t+8 e^{3 t}+1\right )\right ) \\ \end{align*}