83.9.5 problem 5

Internal problem ID [21963]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. X at page 57
Problem number : 5
Date solved : Thursday, October 02, 2025 at 08:20:13 PM
CAS classification : [_quadrature]

\begin{align*} R q^{\prime }+\frac {q}{c}&=E \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 21

ode:=R*diff(q(t),t)+1/c*q(t) = E; 
ic:=[q(0) = 0]; 
dsolve([ode,op(ic)],q(t), singsol=all);
 

\[ q = E c \left (1-{\mathrm e}^{-\frac {t}{c R}}\right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 23

ode=R*D[q[t],t]+1/c*q[t]==e; 
ic={q[0]==0}; 
DSolve[{ode,ic},q[t],t,IncludeSingularSolutions->True]
 

\begin{align*} q(t)&\to c e \left (1-e^{-\frac {t}{c R}}\right ) \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
R = symbols("R") 
E = symbols("E") 
c = symbols("c") 
q = Function("q") 
ode = Eq(R*Derivative(q(t), t) - E + q(t)/c,0) 
ics = {q(0): 0} 
dsolve(ode,func=q(t),ics=ics)
 

\[ q{\left (t \right )} = e c - e c e^{- \frac {t}{R c}} \]