70.16.2 problem 16

Internal problem ID [18965]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.6 (Forced vibrations, Frequency response, and Resonance). Problems at page 272
Problem number : 16
Date solved : Thursday, October 02, 2025 at 03:36:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+2 y&=2 \cos \left (w t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 87

ode:=diff(diff(y(t),t),t)+1/4*diff(y(t),t)+2*y(t) = 2*cos(w*t); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\frac {256 \,{\mathrm e}^{-\frac {t}{8}} \sqrt {127}\, \left (w^{4}-\frac {65}{16} w^{2}+\frac {15}{4}\right ) \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127}+32 \,{\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {\sqrt {127}\, t}{8}\right ) \left (w^{2}-2\right )-32 \cos \left (w t \right ) w^{2}+8 \sin \left (w t \right ) w +64 \cos \left (w t \right )}{16 w^{4}-63 w^{2}+64} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 141

ode=D[y[t],{t,2}]+1/4*D[y[t],t]+2*y[t]==2*Cos[w*t]; 
ic={y[0]==0,Derivative[1][y][0] == 2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {8 e^{-t/8} \left (32 \sqrt {127} w^4 \sin \left (\frac {\sqrt {127} t}{8}\right )-130 \sqrt {127} w^2 \sin \left (\frac {\sqrt {127} t}{8}\right )+508 \left (w^2-2\right ) \cos \left (\frac {\sqrt {127} t}{8}\right )-508 e^{t/8} \left (w^2-2\right ) \cos (t w)+127 e^{t/8} w \sin (t w)+120 \sqrt {127} \sin \left (\frac {\sqrt {127} t}{8}\right )\right )}{127 \left (16 w^4-63 w^2+64\right )} \end{align*}

✓ Sympy. Time used: 0.258 (sec). Leaf size: 182

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(2*y(t) - 2*cos(t*w) + Derivative(y(t), t)/4 + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {32 w^{2} \cos {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64} + \frac {8 w \sin {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64} + \left (\left (\frac {32 w^{2}}{16 w^{4} - 63 w^{2} + 64} - \frac {64}{16 w^{4} - 63 w^{2} + 64}\right ) \cos {\left (\frac {\sqrt {127} t}{8} \right )} + \left (\frac {256 \sqrt {127} w^{4}}{2032 w^{4} - 8001 w^{2} + 8128} - \frac {1040 \sqrt {127} w^{2}}{2032 w^{4} - 8001 w^{2} + 8128} + \frac {960 \sqrt {127}}{2032 w^{4} - 8001 w^{2} + 8128}\right ) \sin {\left (\frac {\sqrt {127} t}{8} \right )}\right ) e^{- \frac {t}{8}} + \frac {64 \cos {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64} \]