Internal problem ID [354]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Examples. Page 437
Problem number: Example 1.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=9 x_{1}\relax (t )+4 x_{2}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-6 x_{1}\relax (t )-x_{2}\relax (t )\\ x_{3}^{\prime }\relax (t )&=6 x_{1}\relax (t )+4 x_{2}\relax (t )+3 x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 66
dsolve([diff(x__1(t),t)=9*x__1(t)+4*x__2(t)+0*x__3(t),diff(x__2(t),t)=-6*x__1(t)-1*x__2(t)+0*x__3(t),diff(x__3(t),t)=6*x__1(t)+4*x__2(t)+3*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = c_{3} {\mathrm e}^{5 t}+\frac {2 c_{2} {\mathrm e}^{3 t}}{3}-\frac {2 c_{1} {\mathrm e}^{3 t}}{3} \] \[ x_{2}\relax (t ) = -c_{2} {\mathrm e}^{3 t}-c_{3} {\mathrm e}^{5 t}+c_{1} {\mathrm e}^{3 t} \] \[ x_{3}\relax (t ) = c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{5 t} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 105
DSolve[{x1'[t]==9*x1[t]+4*x2[t]+0*x3[t],x2'[t]==-6*x1[t]-1*x2[t]+0*x3[t],x3'[t]==6*x1[t]+4*x2[t]+3*x[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{4 t} (c_1 \cosh (t)+(5 c_1+4 c_2) \sinh (t)) \\ \text {x2}(t)\to 3 (c_1+c_2) e^{3 t}-(3 c_1+2 c_2) e^{5 t} \\ \text {x3}(t)\to \int _1^t3 x(K[1])dK[1]+\frac {6}{5} c_1 \left (e^{5 t}-1\right )+\frac {4}{5} c_2 \left (e^{5 t}-1\right )+c_3 \\ \end{align*}