72.14.15 problem 20

Internal problem ID [14878]
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 : 20
Date solved : Tuesday, January 28, 2025 at 07:18:03 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 41

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 94

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

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