15.12.22 problem 22

Internal problem ID [3166]
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 : 22
Date solved : Tuesday, March 04, 2025 at 04:04:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\tan \left (\frac {x}{3}\right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 48

ode:=diff(diff(y(x),x),x)+y(x) = tan(1/3*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \sin \left (x \right )+\cos \left (x \right ) c_{1} -22+9 \left (1-4 \cos \left (\frac {x}{3}\right )^{2}\right ) \sin \left (\frac {x}{3}\right ) \ln \left (\sec \left (\frac {x}{3}\right )+\tan \left (\frac {x}{3}\right )\right )+36 \cos \left (\frac {x}{3}\right )^{2} \]

✓ Mathematica. Time used: 0.317 (sec). Leaf size: 36

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

\[ y(x)\to -9 \sin (x) \text {arctanh}\left (\sin \left (\frac {x}{3}\right )\right )+18 \cos \left (\frac {2 x}{3}\right )+c_1 \cos (x)+c_2 \sin (x)-4 \]

✓ Sympy. Time used: 1.227 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (C_{1} - \int \sin {\left (x \right )} \tan ^{2}{\left (\frac {x}{3} \right )}\, dx\right ) \cos {\left (x \right )} + \left (C_{2} + \int \cos {\left (x \right )} \tan ^{2}{\left (\frac {x}{3} \right )}\, dx\right ) \sin {\left (x \right )} \]