2.13.5 problem problem 11

Internal problem ID [926]
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
Date solved : Tuesday, September 30, 2025 at 04:19:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=4 x_{1} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.170 (sec). Leaf size: 169

ode:=[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)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}+c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-c_3 \sin \left (24^{{1}/{4}} t \right )+c_4 \cos \left (24^{{1}/{4}} t \right ) \\ x_{2} \left (t \right ) &= -24^{{1}/{4}} \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}-c_2 \,{\mathrm e}^{24^{{1}/{4}} t}+\cos \left (24^{{1}/{4}} t \right ) c_3 +\sin \left (24^{{1}/{4}} t \right ) c_4 \right ) \\ x_{3} \left (t \right ) &= \sqrt {6}\, \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}+c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-c_4 \cos \left (24^{{1}/{4}} t \right )+c_3 \sin \left (24^{{1}/{4}} t \right )\right ) \\ x_{4} \left (t \right ) &= -\frac {24^{{3}/{4}} \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}-c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-\cos \left (24^{{1}/{4}} t \right ) c_3 -\sin \left (24^{{1}/{4}} t \right ) c_4 \right )}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 400

ode={D[ x1[t],t]==x2[t],D[ x2[t],t]==2*x3[t],D[ x3[t],t]==3*x4[t],D[ x4[t],t]==4*x1[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{4} c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+\frac {3}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+\frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ]\\ \text {x2}(t)&\to \frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+6 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+\frac {3}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ]\\ \text {x3}(t)&\to \frac {1}{4} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {3}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+3 c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+3 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ]\\ \text {x4}(t)&\to \frac {1}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 272

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__4(t) + Derivative(x__3(t), t),0),Eq(-4*x__1(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {2^{\frac {3}{4}} \sqrt [4]{3} C_{1} \sin {\left (\sqrt [4]{24} t \right )}}{4} - \frac {2^{\frac {3}{4}} \sqrt [4]{3} C_{2} \cos {\left (\sqrt [4]{24} t \right )}}{4} - \frac {2^{\frac {3}{4}} \sqrt [4]{3} C_{3} e^{- \sqrt [4]{24} t}}{4} + \frac {2^{\frac {3}{4}} \sqrt [4]{3} C_{4} e^{\sqrt [4]{24} t}}{4}, \ x^{2}{\left (t \right )} = - \frac {\sqrt {6} C_{1} \cos {\left (\sqrt [4]{24} t \right )}}{2} + \frac {\sqrt {6} C_{2} \sin {\left (\sqrt [4]{24} t \right )}}{2} + \frac {\sqrt {6} C_{3} e^{- \sqrt [4]{24} t}}{2} + \frac {\sqrt {6} C_{4} e^{\sqrt [4]{24} t}}{2}, \ x^{3}{\left (t \right )} = \frac {\sqrt [4]{54} C_{1} \sin {\left (\sqrt [4]{24} t \right )}}{2} + \frac {\sqrt [4]{54} C_{2} \cos {\left (\sqrt [4]{24} t \right )}}{2} - \frac {\sqrt [4]{54} C_{3} e^{- \sqrt [4]{24} t}}{2} + \frac {\sqrt [4]{54} C_{4} e^{\sqrt [4]{24} t}}{2}, \ x^{4}{\left (t \right )} = C_{1} \cos {\left (\sqrt [4]{24} t \right )} - C_{2} \sin {\left (\sqrt [4]{24} t \right )} + C_{3} e^{- \sqrt [4]{24} t} + C_{4} e^{\sqrt [4]{24} t}\right ] \]