78.14.6 problem 3 (d)

Internal problem ID [18271]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 19. The Method of Variation of Parameters. Problems at page 135
Problem number : 3 (d)
Date solved : Monday, March 31, 2025 at 05:24:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&={\mathrm e}^{-x} \sec \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+5*y(x) = exp(-x)*sec(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-\frac {\ln \left (\sec \left (2 x \right )\right ) \cos \left (2 x \right )}{2}+2 c_1 \cos \left (2 x \right )+\sin \left (2 x \right ) \left (x +2 c_2 \right )\right ) {\mathrm e}^{-x}}{2} \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 42

ode=D[y[x],{x,2}] +2*D[y[x],x]+5*y[x]==Exp[-x]*Sec[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^{-x} (2 (x+2 c_1) \sin (2 x)+\cos (2 x) (\log (\cos (2 x))+4 c_2)) \]

✓ Sympy. Time used: 0.558 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x)/cos(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (C_{1} + \frac {x}{2}\right ) \sin {\left (2 x \right )} + \left (C_{2} + \frac {\log {\left (\cos {\left (2 x \right )} \right )}}{4}\right ) \cos {\left (2 x \right )}\right ) e^{- x} \]