62.26.2 problem Ex 2

Internal problem ID [12859]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 49. Variation of parameters. Page 106
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:48:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 23

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

\[ y = \sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} -\cos \left (x \right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 41

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

\[ y(x)\to \sin (x) \int _1^x\sin (K[1])dK[1]+\cos (x) (-\text {arctanh}(\sin (x)))+\sin (x) \cos (x)+c_1 \cos (x)+c_2 \sin (x) \]

✓ Sympy. Time used: 0.325 (sec). Leaf size: 29

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

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