7.25.27 problem 38

Internal problem ID [647]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 38
Date solved : Saturday, March 29, 2025 at 05:01:31 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.190 (sec). Leaf size: 74

ode:=[diff(x__1(t),t) = x__1(t), diff(x__2(t),t) = 2*x__1(t)+2*x__2(t), diff(x__3(t),t) = 3*x__2(t)+3*x__3(t), diff(x__4(t),t) = 4*x__3(t)+4*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 128

ode={D[x1[t],t]==x1[t],D[x2[t],t]==2*x1[t]+2*x2[t],D[x3[t],t]==3*x2[t]+3*x3[t],D[x4[t],t]==4*x3[t]+4*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{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 \left (e^t-1\right )+c_4 e^t\right )\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 83

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__1(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 2*x__2(t) + Derivative(x__2(t), t),0),Eq(-3*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0),Eq(-4*x__3(t) - 4*x__4(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)
 

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