26.9.11 problem Exercise 22.11, page 240

Internal problem ID [7120]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22.11, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to \cos (x) \int _1^x-\sin (K[1]) \tan ^2(K[1])dK[1]+\sin (x) \text {arctanh}(\sin (x))-\sin ^2(x)+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.235 (sec). Leaf size: 31

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

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