72.14.4 problem 6

Internal problem ID [14867]
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 : 6
Date solved : Tuesday, January 28, 2025 at 07:17:54 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 50

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

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 108

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

\begin{align*} x(t)\to e^t (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ z(t)\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ y(t)\to c_3 e^{-t} \\ x(t)\to e^t (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ z(t)\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ y(t)\to 0 \\ \end{align*}