81.18.3 problem 22-3

Internal problem ID [21742]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 22. Electrical Circuits
Problem number : 22-3
Date solved : Thursday, October 02, 2025 at 08:01:42 PM
CAS classification : [_quadrature]

\begin{align*} L i^{\prime }+R i&=E_{0} \end{align*}

With initial conditions

\begin{align*} i \left (0\right )&=i_{0} \\ \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 26

ode:=L*diff(i(t),t)+R*i(t) = E__0; 
ic:=[i(0) = i__0]; 
dsolve([ode,op(ic)],i(t), singsol=all);
 

\[ i = \frac {\left (i_{0} R -E_{0} \right ) {\mathrm e}^{-\frac {R t}{L}}+E_{0}}{R} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 34

ode=L*D[i[t],t]+R*i[t]==e0; 
ic={i[0]==i0}; 
DSolve[{ode,ic},i[t],t,IncludeSingularSolutions->True]
 

\begin{align*} i(t)&\to \frac {e^{-\frac {R t}{L}} \left (\text {e0} \left (e^{\frac {R t}{L}}-1\right )+\text {i0} R\right )}{R} \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
i0 = symbols("i0") 
R = symbols("R") 
e0 = symbols("e0") 
i = Function("i") 
ode = Eq(L*Derivative(i(t), t) + R*i(t) - e0,0) 
ics = {i(0): i0} 
dsolve(ode,func=i(t),ics=ics)
 

\[ i{\left (t \right )} = \frac {e_{0}}{R} + \frac {\left (R i_{0} - e_{0}\right ) e^{- \frac {R t}{L}}}{R} \]