50.24.6 problem 7(c)

Internal problem ID [8172]
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 : Monday, January 27, 2025 at 03:45:12 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*}

✓ Solution by Maple

Time used: 0.654 (sec). Leaf size: 47

dsolve([L*diff(i(t),t)+R*i(t)=E__0*sin(omega*t),i(0) = 0],i(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.086 (sec). Leaf size: 47

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

\[ i(t)\to \frac {\text {E0} \left (L \omega e^{-\frac {R t}{L}}-L \omega \cos (t \omega )+R \sin (t \omega )\right )}{L^2 \omega ^2+R^2} \]