50.24.5 problem 7(b)

Internal problem ID [8171]
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(b)
Date solved : Sunday, March 30, 2025 at 12:47:17 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 17

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

\[ i = \frac {E_{0} {\mathrm e}^{-\frac {R t}{L}}}{L} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 26

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

\[ i(t)\to \frac {\text {e0} (\theta (t)-\theta (0)) e^{-\frac {R t}{L}}}{L} \]

✓ Sympy. Time used: 1.087 (sec). Leaf size: 58

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

\[ \frac {E^{0} \int \operatorname {Dirac}{\left (t \right )} e^{\frac {R t}{L}}\, dt - R \int i{\left (t \right )} e^{\frac {R t}{L}}\, dt}{L} = \frac {E^{0} \int \limits ^{0} \operatorname {Dirac}{\left (t \right )} e^{\frac {R t}{L}}\, dt}{L} - \frac {R \int \limits ^{0} i{\left (t \right )} e^{\frac {R t}{L}}\, dt}{L} \]