62.29.9 problem Ex 10

Internal problem ID [12889]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number : Ex 10
Date solved : Monday, March 31, 2025 at 07:23:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sec \left (x \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 42

ode:=diff(diff(y(x),x),x)+4*y(x) = sec(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (1-2 \cos \left (x \right )^{2}\right ) \ln \left (\sec \left (x \right )\right )+2 c_1 \cos \left (x \right )^{2}+2 \sin \left (x \right ) \left (x +c_2 \right ) \cos \left (x \right )-\sin \left (x \right )^{2}-c_1 \]

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 49

ode=D[y[x],{x,2}]+4*y[x]==Sec[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \sin (2 x) \int _1^x\frac {1}{2} \left (1-\tan ^2(K[1])\right )dK[1]+c_2 \sin (2 x)+\cos (2 x) (\log (\cos (x))+c_1) \]

✓ Sympy. Time used: 0.333 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + Derivative(y(x), (x, 2)) - 1/cos(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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