72.14.13 problem 17

Internal problem ID [14876]
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 : 17
Date solved : Tuesday, January 28, 2025 at 07:18:01 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+3 y\\ y^{\prime }&=-y+z \left (t \right )\\ z^{\prime }\left (t \right )&=5 x \left (t \right )-5 y \end{align*}

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 100

dsolve([diff(x(t),t)=-4*x(t)+3*y(t)+0*z(t),diff(y(t),t)=0*x(t)-1*y(t)+1*z(t),diff(z(t),t)=5*x(t)-5*y(t)+0*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\frac {6 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2}}{5}-\frac {3 \,{\mathrm e}^{-2 t} \cos \left (t \right ) c_{2}}{5}+\frac {6 c_{3} {\mathrm e}^{-2 t} \cos \left (t \right )}{5}+\frac {3 c_{3} {\mathrm e}^{-2 t} \sin \left (t \right )}{5} \\ y &= {\mathrm e}^{-t} c_{1} +{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2} +c_{3} {\mathrm e}^{-2 t} \cos \left (t \right ) \\ z &= -{\mathrm e}^{-2 t} \left (c_{2} \sin \left (t \right )+\sin \left (t \right ) c_{3} -c_{2} \cos \left (t \right )+\cos \left (t \right ) c_{3} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 152

DSolve[{D[x[t],t]==-4*x[t]+3*y[t]+0*z[t],D[y[t],t]==0*x[t]-1*y[t]+1*z[t],D[z[t],t]==5*x[t]-5*y[t]+0*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t-3 (c_1-c_2+c_3) \cos (t)-3 (3 c_1-3 c_2+c_3) \sin (t)\right ) \\ y(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t+(-5 c_1+5 c_2-3 c_3) \cos (t)-(5 c_1-5 c_2+c_3) \sin (t)\right ) \\ z(t)\to e^{-2 t} (c_3 \cos (t)+(5 c_1-5 c_2+2 c_3) \sin (t)) \\ \end{align*}