52.10.37 problem 40

Internal problem ID [8431]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 40
Date solved : Monday, January 27, 2025 at 04:00:26 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y+2 z \left (t \right )\\ y^{\prime }&=3 x \left (t \right )+6 z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x \left (t \right )-3 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 91

dsolve([diff(x(t),t)=2*x(t)+y(t)+2*z(t),diff(y(t),t)=3*x(t)+6*z(t),diff(z(t),t)=-4*x(t)-3*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (\cos \left (2 t \right ) c_{2} +2 \cos \left (2 t \right ) c_3 +2 \sin \left (2 t \right ) c_{2} -\sin \left (2 t \right ) c_3 \right )}{2} \\ y &= -2 \,{\mathrm e}^{-3 t} c_{1} -\frac {3 c_{2} {\mathrm e}^{t} \cos \left (2 t \right )}{2}+\frac {3 c_3 \,{\mathrm e}^{t} \sin \left (2 t \right )}{2} \\ z \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{t} \sin \left (2 t \right )+c_3 \,{\mathrm e}^{t} \cos \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 176

DSolve[{D[x[t],t]==2*x[t]+y[t]+2*z[t],D[y[t],t]==3*x[t]+6*z[t],D[z[t],t]==-4*x[t]-3*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^t (2 c_1 \cos (2 t)+(c_1+c_2+2 c_3) \sin (2 t)) \\ y(t)\to \frac {2}{5} (-3 c_1+c_2-3 c_3) e^{-3 t}+\frac {3}{5} (2 c_1+c_2+2 c_3) e^t \cos (2 t)-\frac {3}{5} (3 c_1-c_2-2 c_3) e^t \sin (t) \cos (t) \\ z(t)\to \frac {1}{5} e^{-3 t} \left (-(3 c_1-c_2-2 c_3) e^{4 t} \cos (2 t)-2 (2 c_1+c_2+2 c_3) e^{4 t} \sin (2 t)+3 c_1-c_2+3 c_3\right ) \\ \end{align*}