Internal problem ID [11035]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems
page 514
Problem number: Problem 4(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y \left (t \right )+3 z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right )+2 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+2 y \left (t \right )-2 z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 91
dsolve([diff(x(t),t)=3*x(t)-2*y(t)+3*z(t),diff(y(t),t)=x(t)-y(t)+2*z(t)+2*exp(-t),diff(z(t),t)=-2*x(t)+2*y(t)-2*z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -{\mathrm e}^{t} c_{1} -c_{2} {\mathrm e}^{-2 t}-c_{3} {\mathrm e}^{t} t -\frac {3 \,{\mathrm e}^{t} c_{3}}{2}+2 \,{\mathrm e}^{-t} \] \[ y \left (t \right ) = {\mathrm e}^{-t}+\frac {{\mathrm e}^{t} c_{1}}{2}-c_{2} {\mathrm e}^{-2 t}+\frac {c_{3} {\mathrm e}^{t} t}{2}-{\mathrm e}^{t} c_{3} \] \[ z \left (t \right ) = -2 \,{\mathrm e}^{-t}+{\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t} t \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 174
DSolve[{x'[t]==3*x[t]-2*y[t]+3*z[t],y'[t]==x[t]-y[t]+2*z[t]+2*Exp[-t],z'[t]==-2*x[t]+2*y[t]-2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{9} e^{-2 t} \left (18 e^t+e^{3 t} (c_1 (6 t+13)+c_3 (6 t+7)-6 c_2)-4 c_1+6 c_2-7 c_3\right ) \\ y(t)\to \frac {1}{9} e^{-2 t} \left (9 e^t+e^{3 t} (c_1 (4-3 t)+c_3 (7-3 t)+3 c_2)-4 c_1+6 c_2-7 c_3\right ) \\ z(t)\to \frac {1}{9} e^{-2 t} \left (-18 e^t+2 e^{3 t} (-(c_1 (3 t+2))-3 c_3 t+3 c_2+c_3)+4 c_1-6 c_2+7 c_3\right ) \\ \end{align*}