60.9.45 problem 1900

Internal problem ID [11899]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1900
Date solved : Monday, January 27, 2025 at 11:44:16 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )-4 y \left (t \right )+z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{4 t} \\ y \left (t \right ) &= \frac {c_3 \,{\mathrm e}^{4 t}}{6}+{\mathrm e}^{-2 t} c_{2} \\ z &= \frac {c_3 \,{\mathrm e}^{4 t}}{9}+c_{1} {\mathrm e}^{t}+\frac {4 \,{\mathrm e}^{-2 t} c_{2}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 88

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

\begin{align*} x(t)\to c_1 e^{4 t} \\ y(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (e^{6 t}-1\right )+6 c_2\right ) \\ z(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (e^{3 t}+e^{6 t}-2\right )-12 c_2 \left (e^{3 t}-1\right )+9 c_3 e^{3 t}\right ) \\ \end{align*}