10.17.9 problem 9

Internal problem ID [1424]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 9
Date solved : Saturday, March 29, 2025 at 10:54:49 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 3\\ x_{2} \left (0\right ) = -2 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 28

ode:=[diff(x__1(t),t) = 2*x__1(t)+3/2*x__2(t), diff(x__2(t),t) = -3/2*x__1(t)-x__2(t)]; 
ic:=x__1(0) = 3x__2(0) = -2; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {3 t}{2}+3\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {9 t}{2}+6\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode={D[ x1[t],t]==2*x1[t]+3/2*x2[t],D[ x2[t],t]==-3/2*x1[t]-1*x2[t]}; 
ic={x1[0]==3,x2[0]==-2}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {3}{2} e^{t/2} (t+2) \\ \text {x2}(t)\to -\frac {1}{2} e^{t/2} (3 t+4) \\ \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 51

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

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