76.8.21 problem 21

Internal problem ID [17431]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 21
Date solved : Thursday, March 13, 2025 at 10:08:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}i \left (t \right )&=\frac {i \left (t \right )}{2}-\frac {v \left (t \right )}{8}\\ \frac {d}{d t}v \left (t \right )&=2 i \left (t \right )-\frac {v \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 23

ode:=[diff(i(t),t) = 1/2*i(t)-1/8*v(t), diff(v(t),t) = 2*i(t)-1/2*v(t)]; 
dsolve(ode);
 

\begin{align*} i &= c_{1} t +c_{2} \\ v &= 4 c_{1} t -8 c_{1} +4 c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 40

ode={D[i[t],t]==1/2*i[t]-1/8*v[t],D[v[t],t]==2*i[t]-1/2*v[t]}; 
ic={}; 
DSolve[{ode,ic},{i[t],v[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} i(t)\to \frac {1}{2} c_1 (t+2)-\frac {c_2 t}{8} \\ v(t)\to 2 c_1 t-\frac {c_2 t}{2}+c_2 \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 22

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

\[ \left [ i{\left (t \right )} = \frac {C_{1}}{2} + \frac {C_{2} t}{2} + C_{2}, \ v{\left (t \right )} = 2 C_{1} + 2 C_{2} t\right ] \]