44.24.6 problem 7(c)

Internal problem ID [9458]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 The Unit Step and Impulse Functions. Page 303
Problem number : 7(c)
Date solved : Tuesday, September 30, 2025 at 06:19:01 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} L i^{\prime }+R i&=E_{0} \sin \left (\omega t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.099 (sec). Leaf size: 44

ode:=L*diff(i(t),t)+R*i(t) = E__0*sin(omega*t); 
ic:=[i(0) = 0]; 
dsolve([ode,op(ic)],i(t),method='laplace');
 

\[ i = \frac {E_{0} \left (\sin \left (\omega t \right ) R +L \omega \left (-\cos \left (\omega t \right )+{\mathrm e}^{-\frac {R t}{L}}\right )\right )}{\omega ^{2} L^{2}+R^{2}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 41

ode=L*D[i[t],t]+R*i[t]==E0*Sin[\[Omega]*t]; 
ic={i[0]==0}; 
DSolve[{ode,ic},i[t],t,IncludeSingularSolutions->True]
 

\begin{align*} i(t)&\to e^{-\frac {R t}{L}} \int _0^t\frac {e^{\frac {R K[1]}{L}} \text {E0} \sin (\omega K[1])}{L}dK[1] \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
E__0 = symbols("E__0") 
L = symbols("L") 
R = symbols("R") 
omega = symbols("omega") 
i = Function("i") 
ode = Eq(-E__0*sin(omega*t) + L*Derivative(i(t), t) + R*i(t),0) 
ics = {i(0): 0} 
dsolve(ode,func=i(t),ics=ics)
 

\[ i{\left (t \right )} = - \frac {E^{0} L \omega \cos {\left (\omega t \right )}}{L^{2} \omega ^{2} + R^{2}} + \frac {E^{0} L \omega e^{- \frac {R t}{L}}}{L^{2} \omega ^{2} + R^{2}} + \frac {E^{0} R \sin {\left (\omega t \right )}}{L^{2} \omega ^{2} + R^{2}} \]