87.19.13 problem 23

Internal problem ID [23683]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 149
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:44:13 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )-x_{3} \left (t \right )+x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{2} \left (t \right )+x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{3} \left (t \right )-x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=2 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 66

ode:=[diff(x__1(t),t) = x__2(t)-x__3(t)+x__4(t), diff(x__2(t),t) = -x__2(t)+x__4(t), diff(x__3(t),t) = x__3(t)-x__4(t), diff(x__4(t),t) = 2*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {7 c_4 \,{\mathrm e}^{2 t}}{6}-{\mathrm e}^{t} c_2 -{\mathrm e}^{-t} c_3 +c_1 \\ x_{2} \left (t \right ) &= \frac {c_4 \,{\mathrm e}^{2 t}}{3}+{\mathrm e}^{-t} c_3 \\ x_{3} \left (t \right ) &= -c_4 \,{\mathrm e}^{2 t}+{\mathrm e}^{t} c_2 \\ x_{4} \left (t \right ) &= c_4 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 106

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

\begin{align*} \text {x1}(t)&\to \frac {1}{2} e^{-t} \left (2 (c_1+c_2+c_3-c_4) e^t+e^{2 t} (2 c_4 t-2 c_3+c_4)-2 c_2+c_4\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{-t} \left (c_4 \left (e^{2 t}-1\right )+2 c_2\right )\\ \text {x3}(t)&\to e^t (c_3-c_4 t)\\ \text {x4}(t)&\to c_4 e^t \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
x4 = Function("x4") 
ode=[Eq(-x2(t) + x3(t) - x4(t) + Derivative(x1(t), t),0),Eq(x2(t) - x4(t) + Derivative(x2(t), t),0),Eq(-x3(t) + x4(t) + Derivative(x3(t), t),0),Eq(-x4(t) + Derivative(x4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t),x3(t),x4(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = C_{1} + C_{2} t e^{t} - C_{3} e^{- t} + \left (\frac {C_{2}}{2} + C_{4}\right ) e^{t}, \ x_{2}{\left (t \right )} = \frac {C_{2} e^{t}}{2} + C_{3} e^{- t}, \ x_{3}{\left (t \right )} = - C_{2} t e^{t} - C_{4} e^{t}, \ x_{4}{\left (t \right )} = C_{2} e^{t}\right ] \]