34.4.20 problem 22 (b)

Internal problem ID [7947]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary problems. Page 39
Problem number : 22 (b)
Date solved : Tuesday, September 30, 2025 at 05:11:49 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} L i^{\prime }+R i&=E \sin \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 43

ode:=L*diff(i(t),t)+R*i(t) = E*sin(2*t); 
ic:=[i(0) = 0]; 
dsolve([ode,op(ic)],i(t), singsol=all);
 

\[ i = -\frac {2 E \left (L \cos \left (2 t \right )-L \,{\mathrm e}^{-\frac {R t}{L}}-\frac {\sin \left (2 t \right ) R}{2}\right )}{4 L^{2}+R^{2}} \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 41

ode=L*D[i[t],t]+R*i[t]==e*Sin[2*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 e^{\frac {R K[1]}{L}} \sin (2 K[1])}{L}dK[1] \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
E = symbols("E") 
L = symbols("L") 
R = symbols("R") 
i = Function("i") 
ode = Eq(L*Derivative(i(t), t) + R*i(t) - E*sin(2*t),0) 
ics = {i(0): 0} 
dsolve(ode,func=i(t),ics=ics)
 

\[ i{\left (t \right )} = - \frac {2 e L \cos {\left (2 t \right )}}{4 L^{2} + R^{2}} + \frac {2 e L e^{- \frac {R t}{L}}}{4 L^{2} + R^{2}} + \frac {e R \sin {\left (2 t \right )}}{4 L^{2} + R^{2}} \]