72.14.5 problem 7

Internal problem ID [14868]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 07:17:55 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 37

dsolve([diff(x(t),t)=1*x(t)+0*y(t)+0*z(t),diff(y(t),t)=0*x(t)+2*y(t)-1*z(t),diff(z(t),t)=0*x(t)-1*y(t)+2*z(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 144

DSolve[{D[x[t],t]==1*x[t]+0*y[t]+0*z[t],D[y[t],t]==0*x[t]+2*y[t]-1*z[t],D[z[t],t]==0*x[t]-1*y[t]+2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 e^t \\ y(t)\to \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ x(t)\to 0 \\ y(t)\to \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ \end{align*}