Internal problem ID [276]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 7.2, Matrices and Linear systems. Page 417
Problem number: problem 11.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{2}\relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=4 x_{1}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 171
dsolve([diff(x__1(t),t)=x__2(t),diff(x__2(t),t)=2*x__3(t),diff(x__3(t),t)=3*x__4(t),diff(x__4(t),t)=4*x__1(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = -\frac {24^{\frac {1}{4}} \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}-c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}+\cos \left (24^{\frac {1}{4}} t \right ) c_{3}+\sin \left (24^{\frac {1}{4}} t \right ) c_{4}\right )}{4} \] \[ x_{2}\relax (t ) = \frac {\sqrt {6}\, \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}+c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-c_{4} \cos \left (24^{\frac {1}{4}} t \right )+c_{3} \sin \left (24^{\frac {1}{4}} t \right )\right )}{2} \] \[ x_{3}\relax (t ) = -\frac {24^{\frac {3}{4}} \left (c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}-c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-\cos \left (24^{\frac {1}{4}} t \right ) c_{3}-\sin \left (24^{\frac {1}{4}} t \right ) c_{4}\right )}{8} \] \[ x_{4}\relax (t ) = c_{1} {\mathrm e}^{-24^{\frac {1}{4}} t}+c_{2} {\mathrm e}^{24^{\frac {1}{4}} t}-c_{3} \sin \left (24^{\frac {1}{4}} t \right )+c_{4} \cos \left (24^{\frac {1}{4}} t \right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 391
DSolve[{x1'[t]==x2[t],x2'[t]==2*x3[t],x3'[t]==3*x4[t],x4'[t]==4*x1[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{4} \left (c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+6 c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ]\right ) \\ \text {x2}(t)\to \frac {1}{4} \left (c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+24 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+6 c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ]\right ) \\ \text {x3}(t)\to \frac {1}{4} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {3}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+3 c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+3 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\ \text {x4}(t)\to \frac {1}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\ \end{align*}