78.14.3 problem 3 (a)

Internal problem ID [18268]
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 (a)
Date solved : Monday, March 31, 2025 at 05:24:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\tan \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+4*y(x) = tan(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (2 x \right ) c_2 +\cos \left (2 x \right ) c_1 -\frac {\cos \left (2 x \right ) \ln \left (\sec \left (2 x \right )+\tan \left (2 x \right )\right )}{4} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 40

ode=D[y[x],{x,2}] +4*y[x]==Tan[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.426 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - tan(2*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} \sin {\left (2 x \right )} + \left (C_{1} + \frac {\log {\left (\sin {\left (2 x \right )} - 1 \right )}}{8} - \frac {\log {\left (\sin {\left (2 x \right )} + 1 \right )}}{8}\right ) \cos {\left (2 x \right )} \]