10.19.15 problem 15

Internal problem ID [1456]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 15
Date solved : Tuesday, March 04, 2025 at 12:36:14 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )-1\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+5 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)-1, diff(x__2(t),t) = 2*x__1(t)-x__2(t)+5]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -2+{\mathrm e}^{-t} \left (\cos \left (\sqrt {2}\, t \right ) c_1 +c_2 \sin \left (\sqrt {2}\, t \right )\right ) \\ x_{2} \left (t \right ) &= 1-\sqrt {2}\, {\mathrm e}^{-t} \left (c_2 \cos \left (\sqrt {2}\, t \right )-c_1 \sin \left (\sqrt {2}\, t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.344 (sec). Leaf size: 85

ode={D[ x1[t],t]==-1*x1[t]-1*x2[t]-1,D[ x2[t],t]==2*x1[t]-1*x2[t]+5}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{-t} \cos \left (\sqrt {2} t\right )-\frac {c_2 e^{-t} \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-2 \\ \text {x2}(t)\to e^{-t} \left (e^t+c_2 \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} t\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.353 (sec). Leaf size: 117

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) + x__2(t) + Derivative(x__1(t), t) + 1,0),Eq(-2*x__1(t) + x__2(t) + Derivative(x__2(t), t) - 5,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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