10.15.1 problem 30

Internal problem ID [1400]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.5, Homogeneous Linear Systems with Constant Coefficients. page 407
Problem number : 30
Date solved : Tuesday, March 04, 2025 at 12:35:12 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{10}+\frac {3 x_{2} \left (t \right )}{40}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{1} \left (t \right )}{10}-\frac {x_{2} \left (t \right )}{5} \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -17\\ x_{2} \left (0\right ) = -21 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = -1/10*x__1(t)+3/40*x__2(t), diff(x__2(t),t) = 1/10*x__1(t)-1/5*x__2(t)]; 
ic:=x__1(0) = -17x__2(0) = -21; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {165 \,{\mathrm e}^{-\frac {t}{20}}}{8}+\frac {29 \,{\mathrm e}^{-\frac {t}{4}}}{8} \\ x_{2} \left (t \right ) &= -\frac {55 \,{\mathrm e}^{-\frac {t}{20}}}{4}-\frac {29 \,{\mathrm e}^{-\frac {t}{4}}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

ode={D[ x1[t],t]==-1/10*x1[t]+3/40*x2[t],D[ x2[t],t]==1/10*x1[t]-1/5*x2[t]}; 
ic={x1[0]==-17,x2[0]==-21}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{8} e^{-t/4} \left (29-165 e^{t/5}\right ) \\ \text {x2}(t)\to -\frac {1}{4} e^{-t/4} \left (55 e^{t/5}+29\right ) \\ \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t)/10 - 3*x__2(t)/40 + Derivative(x__1(t), t),0),Eq(-x__1(t)/10 + x__2(t)/5 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- \frac {t}{4}}}{2} + \frac {3 C_{2} e^{- \frac {t}{20}}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {t}{4}} + C_{2} e^{- \frac {t}{20}}\right ] \]