67.7.8 problem Problem 4(d)

Internal problem ID [14119]
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)
Date solved : Tuesday, January 28, 2025 at 06:14:46 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y+3 z \left (t \right )\\ y^{\prime }&=x \left (t \right )-y+2 z \left (t \right )+2 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+2 y-2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 89

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)],singsol=all)
 

\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{t} t \\ y &= {\mathrm e}^{-t}-\frac {c_{1} {\mathrm e}^{t}}{2}+{\mathrm e}^{-2 t} c_{2} -\frac {c_{3} {\mathrm e}^{t} t}{2}+\frac {7 c_{3} {\mathrm e}^{t}}{4} \\ z \left (t \right ) &= -2 \,{\mathrm e}^{-t}-c_{1} {\mathrm e}^{t}-{\mathrm e}^{-2 t} c_{2} -c_{3} {\mathrm e}^{t} t +\frac {3 c_{3} {\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.256 (sec). Leaf size: 174

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