56.1.76 problem 76

Internal problem ID [8788]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 76
Date solved : Monday, January 27, 2025 at 04:58:38 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 107

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

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