89.26.12 problem 14

Internal problem ID [24879]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Miscellaneous Exercises at page 177
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:49:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\sec \left (x \right )^{3} \tan \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = sec(x)^3*tan(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right ) c_2 +\cos \left (x \right ) c_1 +\frac {\sin \left (x \right )}{3}+\frac {\sec \left (x \right ) \tan \left (x \right )}{6} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+y[x]== Sec[x]^3*Tan[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos (x)+\frac {1}{2} \tan (x) \sec (x)+\sin (x) \left (-\frac {\tan ^2(x)}{3}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.215 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - tan(x)*sec(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} \cos {\left (x \right )} + \left (C_{1} + \frac {1}{2 \left (\cos {\left (2 x \right )} + 1\right )}\right ) \sin {\left (x \right )} + \frac {\sin {\left (3 x \right )}}{6 \left (\cos {\left (2 x \right )} + 1\right )} \]