14.25.16 problem 18

Internal problem ID [2773]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 18
Date solved : Monday, January 27, 2025 at 06:13:08 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 x_{3} \left (t \right )+2 \,{\mathrm e}^{8 t}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+2 x_{3} \left (t \right )+{\mathrm e}^{8 t}\\ x_{3}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )+2 \,{\mathrm e}^{8 t} \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 86

dsolve([diff(x__1(t),t)=3*x__1(t)+2*x__2(t)+4*x__3(t)+2*exp(8*t),diff(x__2(t),t)=2*x__1(t)+0*x__2(t)+2*x__3(t)+exp(8*t),diff(x__3(t),t)=4*x__1(t)+2*x__2(t)+3*x__3(t)+2*exp(8*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= 2 c_3 \,{\mathrm e}^{-t}+2 c_2 \,{\mathrm e}^{8 t}+2 t \,{\mathrm e}^{8 t}+{\mathrm e}^{-t} c_1 \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{8 t}+t \,{\mathrm e}^{8 t} \\ x_{3} \left (t \right ) &= -\frac {5 c_3 \,{\mathrm e}^{-t}}{2}+2 c_2 \,{\mathrm e}^{8 t}+2 t \,{\mathrm e}^{8 t}-{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 139

DSolve[{D[ x1[t],t]==3*x1[t]+2*x2[t]+4*x3[t]+2*Exp[8*t],D[ x2[t],t]==2*x1[t]+0*x2[t]+2*x3[t]+Exp[8*t],D[ x3[t],t]==4*x1[t]+2*x2[t]+3*x3[t]+2*Exp[8*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-t} \left (2 e^{9 t} (9 t+2 c_1+c_2+2 c_3)+5 c_1-2 (c_2+2 c_3)\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (e^{9 t} (9 t+2 c_1+c_2+2 c_3)-2 (c_1-4 c_2+c_3)\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (2 e^{9 t} (9 t+2 c_1+c_2+2 c_3)-4 c_1-2 c_2+5 c_3\right ) \\ \end{align*}