76.8.21 problem 21

Internal problem ID [17510]
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 : Tuesday, January 28, 2025 at 10:39:43 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*}

✓ Solution by Maple

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

dsolve([diff(i(t),t)=1/2*i(t)-1/8*v(t),diff(v(t),t)=2*i(t)-1/2*v(t)],singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[i[t],t]==1/2*i[t]-1/8*v[t],D[v[t],t]==2*i[t]-1/2*v[t]},{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*}