9.5.3 problem Example 4

Internal problem ID [1006]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Examples. Page 437
Problem number : Example 4
Date solved : Monday, January 27, 2025 at 03:22:53 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-5 x_{1} \left (t \right )-3 x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 74

dsolve([diff(x__1(t),t)=0*x__1(t)+1*x__2(t)+2*x__3(t),diff(x__2(t),t)=-5*x__1(t)-3*x__2(t)-7*x__3(t),diff(x__3(t),t)=1*x__1(t)+0*x__2(t)+0*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 134

DSolve[{D[ x1[t],t]==0*x1[t]+1*x2[t]+2*x3[t],D[ x2[t],t]==-5*x1[t]-3*x2[t]-7*x3[t],D[ x3[t],t]==1*x1[t]+0*x2[t]+0*x[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (-2 t^2+2 t+2\right )-c_2 (t-2) t+c_3 (4-3 t) t\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-\left ((2 c_1+c_2+3 c_3) t^2\right )-2 (5 c_1+2 c_2+7 c_3) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left ((2 c_1+c_2+3 c_3) t^2+2 (c_1+c_3) t+2 c_3\right ) \\ \end{align*}