14.25.8 problem 6

Internal problem ID [2765]
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 : 6
Date solved : Monday, January 27, 2025 at 06:12:59 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 46

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

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (t^{3}+6 t^{2}-6 t \right )}{6} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} t^{2}}{2} \\ x_{3} \left (t \right ) &= \frac {\left (t^{3}+6 t^{2}\right ) {\mathrm e}^{t}}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 50

DSolve[{D[ x1[t],t]==-1*x1[t]-1*x2[t]-2*x3[t]+Exp[t],D[ x2[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[ x3[t],t]==2*x1[t]+1*x2[t]+3*x3[t]},{x1[0]==0,x2[0]==0,x3[0]==0},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -\frac {1}{6} e^t t \left (t^2+6 t-6\right ) \\ \text {x2}(t)\to \frac {e^t t^2}{2} \\ \text {x3}(t)\to \frac {1}{6} e^t t^2 (t+6) \\ \end{align*}