15.17.7 problem 12

Internal problem ID [3243]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 26, page 115
Problem number : 12
Date solved : Monday, January 27, 2025 at 07:27:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=3 y \left (t \right )-2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 51

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

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= \left (2 c_3 t +c_2 \right ) {\mathrm e}^{3 t} \\ z &= \left (\frac {3 \,{\mathrm e}^{5 t} \left (10 c_3 t +5 c_2 -2 c_3 \right )}{25}+c_{1} \right ) {\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 78

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

\begin{align*} x(t)\to c_1 e^{3 t} \\ y(t)\to e^{3 t} (2 c_1 t+c_2) \\ z(t)\to \frac {1}{25} e^{-2 t} \left (6 c_1 \left (e^{5 t} (5 t-1)+1\right )+5 \left (3 c_2 \left (e^{5 t}-1\right )+5 c_3\right )\right ) \\ \end{align*}