43.2.9 problem 5

Internal problem ID [8885]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 1.6 Introduction– Linear equations of First Order. Page 41
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:58:56 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} L y^{\prime }+R y&=E \,{\mathrm e}^{i \omega x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 30

ode:=L*diff(y(x),x)+R*y(x) = E*exp(I*omega*x); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {E \left (-{\mathrm e}^{-\frac {R x}{L}}+{\mathrm e}^{i \omega x}\right )}{i L \omega +R} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 43

ode=L*D[y[x],x]+R*y[x]==E0*Exp[I*\[Omega]*x]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\text {E0} e^{-\frac {R x}{L}} \left (-1+e^{\frac {x (R+i L \omega )}{L}}\right )}{R+i L \omega } \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
E = symbols("E") 
L = symbols("L") 
R = symbols("R") 
omega = symbols("omega") 
y = Function("y") 
ode = Eq(L*Derivative(y(x), x) + R*y(x) - E*exp(omega*x*complex(0, 1)),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{\omega x \operatorname {complex}{\left (0,1 \right )} + 1}}{L \omega \operatorname {complex}{\left (0,1 \right )} + R} - \frac {e e^{- \frac {R x}{L}}}{L \omega \operatorname {complex}{\left (0,1 \right )} + R} \]