14.25.14 problem 16

Internal problem ID [2771]
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 : 16
Date solved : Monday, January 27, 2025 at 06:13:06 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}^{2 t}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-4 x_{3} \left (t \right )+2 \,{\mathrm e}^{2 t}\\ x_{3}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )-4 x_{3} \left (t \right )+{\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 83

dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)-1*x__3(t)+exp(2*t),diff(x__2(t),t)=2*x__1(t)+3*x__2(t)-4*x__3(t)+2*exp(2*t),diff(x__3(t),t)=4*x__1(t)+1*x__2(t)-4*x__3(t)+exp(2*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +c_2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +7 c_2 \,{\mathrm e}^{-3 t}+2 \,{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +11 c_2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 2491

DSolve[{D[ x1[t],t]==1*x1[t]+1*x2[t]-1*x3[t]+Exp[2*t],D[ x2[t],t]==2*x1[t]+3*x2[t]-4*x3[t]+2*Exp[2*t],D[ x3[t],t]==4*x1[t]-1*x2[t]-4*x3[t]+Exp[2*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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