14.25.15 problem 17

Internal problem ID [2772]
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 : 17
Date solved : Monday, January 27, 2025 at 06:13:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 86

dsolve([diff(x__1(t),t)=1*x__1(t)-1*x__2(t)-1*x__3(t)+exp(3*t),diff(x__2(t),t)=1*x__1(t)+3*x__2(t)+1*x__3(t)-1*exp(3*t),diff(x__3(t),t)=-3*x__1(t)+1*x__2(t)-1*x__3(t)-exp(3*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{3 t} t +c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -{\mathrm e}^{3 t} t -c_1 \,{\mathrm e}^{-2 t}-c_3 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= -{\mathrm e}^{3 t} t +4 c_1 \,{\mathrm e}^{-2 t}-c_2 \,{\mathrm e}^{2 t}-c_3 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 142

DSolve[{D[ x1[t],t]==1*x1[t]-1*x2[t]-1*x3[t]+Exp[3*t],D[ x2[t],t]==1*x1[t]+3*x2[t]+1*x3[t]-Exp[3*t],D[ x3[t],t]==-3*x1[t]+1*x2[t]-1*x3[t]-Exp[3*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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