60.9.44 problem 1899

Internal problem ID [11898]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1899
Date solved : Monday, January 27, 2025 at 11:44:15 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.139 (sec). Leaf size: 51

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

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {3 c_3 \,{\mathrm e}^{2 t}}{4}+{\mathrm e}^{-2 t} c_{2} \\ z &= c_{1} {\mathrm e}^{3 t}-\frac {3 c_3 \,{\mathrm e}^{2 t}}{2}-\frac {2 \,{\mathrm e}^{-2 t} c_{2}}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 93

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

\begin{align*} x(t)\to c_1 e^{2 t} \\ y(t)\to \frac {1}{4} e^{-2 t} \left (3 c_1 \left (e^{4 t}-1\right )+4 c_2\right ) \\ z(t)\to \frac {1}{10} e^{-2 t} \left (c_1 \left (-15 e^{4 t}+12 e^{5 t}+3\right )+4 c_2 \left (e^{5 t}-1\right )+10 c_3 e^{5 t}\right ) \\ \end{align*}