85.51.1 problem 2

Internal problem ID [22811]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 194
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:14:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} q \left (0\right )&=0 \\ q^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.332 (sec). Leaf size: 237

ode:=L*diff(diff(q(t),t),t)+R*diff(q(t),t)+1/c*q(t) = E__0*sin(omega*t); 
ic:=[q(0) = 0, D(q)(0) = 0]; 
dsolve([ode,op(ic)],q(t), singsol=all);
 

\[ q = -\frac {E_{0} c \left (\omega \left (\left (\omega ^{2} L^{2}+\frac {R^{2}}{2}\right ) c -L \right ) \left ({\mathrm e}^{\frac {\left (-c R +\sqrt {c^{2} R^{2}-4 L c}\right ) t}{2 L c}}-{\mathrm e}^{-\frac {\left (c R +\sqrt {c^{2} R^{2}-4 L c}\right ) t}{2 L c}}\right ) \sqrt {c^{2} R^{2}-4 L c}-2 \left (-\frac {R^{2} c}{4}+L \right ) \left (\left (-2 L c \,\omega ^{2}+2\right ) \sin \left (\omega t \right )+\omega c R \left ({\mathrm e}^{\frac {\left (-c R +\sqrt {c^{2} R^{2}-4 L c}\right ) t}{2 L c}}+{\mathrm e}^{-\frac {\left (c R +\sqrt {c^{2} R^{2}-4 L c}\right ) t}{2 L c}}-2 \cos \left (\omega t \right )\right )\right )\right )}{4 \left (1+\left (\omega ^{4} L^{2}+R^{2} \omega ^{2}\right ) c^{2}-2 L c \,\omega ^{2}\right ) \left (-\frac {R^{2} c}{4}+L \right )} \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 288

ode=L*D[q[t],{t,2}]+R*D[q[t],t]+1/c*q[t]==e0*Sin[\[Omega]*t]; 
ic={q[0]==0,Derivative[1][q][0] ==0}; 
DSolve[{ode,ic},q[t],t,IncludeSingularSolutions->True]
 

\begin{align*} q(t)&\to -\frac {c \text {e0} \left (-\sqrt {c} \omega e^{-\frac {t \left (\frac {\sqrt {c R^2-4 L}}{\sqrt {c}}+R\right )}{2 L}} \left (2 c L^2 \omega ^2 \left (e^{\frac {t \sqrt {c R^2-4 L}}{\sqrt {c} L}}-1\right )+c R^2 \left (e^{\frac {t \sqrt {c R^2-4 L}}{\sqrt {c} L}}-1\right )+\sqrt {c} R \sqrt {c R^2-4 L} \left (e^{\frac {t \sqrt {c R^2-4 L}}{\sqrt {c} L}}+1\right )-2 L \left (e^{\frac {t \sqrt {c R^2-4 L}}{\sqrt {c} L}}-1\right )\right )+2 \sqrt {c R^2-4 L} \left (c L \omega ^2-1\right ) \sin (t \omega )+2 c R \omega \sqrt {c R^2-4 L} \cos (t \omega )\right )}{2 \sqrt {c R^2-4 L} \left (c^2 \omega ^2 \left (L^2 \omega ^2+R^2\right )-2 c L \omega ^2+1\right )} \end{align*}

✓ Sympy. Time used: 0.534 (sec). Leaf size: 1056

from sympy import * 
t = symbols("t") 
L = symbols("L") 
R = symbols("R") 
c = symbols("c") 
e0 = symbols("e0") 
q = Function("q") 
ode = Eq(L*Derivative(q(t), (t, 2)) + R*Derivative(q(t), t) - e0*sin(t*w) + q(t)/c,0) 
ics = {q(0): 0, Subs(Derivative(q(t), t), t, 0): 0} 
dsolve(ode,func=q(t),ics=ics)
 

\[ \text {Solution too large to show} \]