56.1.71 problem 71

Internal problem ID [8783]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 71
Date solved : Monday, January 27, 2025 at 04:58:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=9 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=6 x \left (t \right )+4 y \left (t \right )+3 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 57

dsolve([diff(x(t),t)=9*x(t)+4*y(t),diff(y(t),t)=-6*x(t)-y(t),diff(z(t),t)=6*x(t)+4*y(t)+3*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{5 t}-\frac {3 c_3 \,{\mathrm e}^{3 t}}{2} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+c_3 \,{\mathrm e}^{3 t}+c_{1} {\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 103

DSolve[{D[x[t],t]==9*x[t]+4*y[t],D[y[t],t]==-6*x[t]-y[t],D[z[t],t]==6*x[t]+4*y[t]+3*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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