67.7.7 problem Problem 4(c)

Internal problem ID [14118]
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(c)
Date solved : Tuesday, January 28, 2025 at 06:14:45 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )-2 y+4 z \left (t \right )\\ y^{\prime }&=-2 x \left (t \right )+y+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x \left (t \right )-2 y+6 z \left (t \right )+{\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.142 (sec). Leaf size: 76

dsolve([diff(x(t),t)=-2*x(t)-2*y(t)+4*z(t),diff(y(t),t)=-2*x(t)+1*y(t)+2*z(t),diff(z(t),t)=-4*x(t)-2*y(t)+6*z(t)+exp(2*t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 118

DSolve[{D[x[t],t]==-2*x[t]-2*y[t]+4*z[t],D[y[t],t]==-2*x[t]+y[t]+2*z[t],D[z[t],t]==-4*x[t]-2*y[t]+6*z[t]+Exp[2*t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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