72.14.10 problem 14

Internal problem ID [14873]
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 : 14
Date solved : Tuesday, January 28, 2025 at 07:17:58 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.152 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 57

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

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