Internal problem ID [341]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number: problem 38.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )+2 x_{2} \relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{2} \relax (t )+3 x_{3} \relax (t )\\ x_{4}^{\prime }\relax (t )&=4 x_{3} \relax (t )+4 x_{4} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 75
dsolve([diff(x__1(t),t)=x__1(t)+0*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=2*x__1(t)+2*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+3*x__2(t)+3*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+4*x__3(t)+4*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \relax (t ) = -\frac {c_{4} {\mathrm e}^{t}}{4} \] \[ x_{2} \relax (t ) = \frac {c_{2} {\mathrm e}^{2 t}}{6}+\frac {c_{4} {\mathrm e}^{t}}{2} \] \[ x_{3} \relax (t ) = -\frac {c_{2} {\mathrm e}^{2 t}}{2}-\frac {c_{3} {\mathrm e}^{3 t}}{4}-\frac {3 c_{4} {\mathrm e}^{t}}{4} \] \[ x_{4} \relax (t ) = c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t}+c_{4} {\mathrm e}^{t} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 128
DSolve[{x1'[t]==1*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],x2'[t]==2*x1[t]+2*x2[t]+0*x3[t]+0*x4[t],x3'[t]==0*x1[t]+3*x2[t]+3*x3[t]+0*x4[t],x4'[t]==0*x1[t]+0*x2[t]+4*x3[t]+4*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to e^t \left (2 c_1 \left (e^t-1\right )+c_2 e^t\right ) \\ \text {x3}(t)\to e^t \left (3 c_1 \left (e^t-1\right )^2+e^t \left (3 c_2 \left (e^t-1\right )+c_3 e^t\right )\right ) \\ \text {x4}(t)\to e^t \left (4 c_1 \left (e^t-1\right )^3+e^t \left (6 c_2 \left (e^t-1\right )^2+e^t \left ((4 c_3+c_4) e^t-4 c_3\right )\right )\right ) \\ \end{align*}