89.26.13 problem 15

Internal problem ID [24880]
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 : 15
Date solved : Thursday, October 02, 2025 at 10:49:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to -\sin (x) \arctan (\tan (x))-\frac {\sec (x)}{2}+\sin (x) \tan (x)+c_2 \sin (x)+\cos (x) (-\log (\cos (x))+c_1) \end{align*}

✓ Sympy. Time used: 0.208 (sec). Leaf size: 29

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

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