10.18.13 problem 13

Internal problem ID [1440]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 12:35:57 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}+\frac {{\mathrm e}^{-\frac {t}{2}}}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 46

ode:=[diff(x__1(t),t) = -1/2*x__1(t)-1/8*x__2(t)+1/2*exp(-1/2*t), diff(x__2(t),t) = 2*x__1(t)-1/2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (c_2 \cos \left (\frac {t}{2}\right )-c_1 \sin \left (\frac {t}{2}\right )\right )}{4} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (4+\cos \left (\frac {t}{2}\right ) c_1 +\sin \left (\frac {t}{2}\right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 69

ode={D[ x1[t],t]==-1/2*x1[t]-1/8*x2[t]+1/2*Exp[-t/2],D[ x2[t],t]==2*x1[t]-1/2*x2[t]+0}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t/2} \left (4 c_1 \cos \left (\frac {t}{2}\right )-c_2 \sin \left (\frac {t}{2}\right )\right ) \\ \text {x2}(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {t}{2}\right )+4 c_1 \sin \left (\frac {t}{2}\right )+4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t)/2 + x__2(t)/8 + Derivative(x__1(t), t) - exp(-t/2)/2,0),Eq(-2*x__1(t) + x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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