15.12.12 problem 12

Internal problem ID [3156]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 20, page 90
Problem number : 12
Date solved : Sunday, March 30, 2025 at 01:19:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 40

ode:=diff(diff(y(x),x),x)+a^2*y(x) = sec(a*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 39

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

\[ y(x)\to \frac {\cos (a x) \left (\log (\cos (a x))+a^2 c_1\right )+a (x+a c_2) \sin (a x)}{a^2} \]

✓ Sympy. Time used: 0.461 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*y(x) + Derivative(y(x), (x, 2)) - 1/cos(a*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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