87.18.22 problem 22

Internal problem ID [23663]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 22
Date solved : Thursday, October 02, 2025 at 09:44:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x)+y(x) = sec(x)^3; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sin \left (x \right )+\frac {\cos \left (x \right )}{2}+\frac {\sec \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 23

ode=D[y[x],{x,2}]+y[x]==Sec[x]^3; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \sec (x) (2 \sin (2 x)+\cos (2 x)+3) \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - sec(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sin {\left (x \right )} + \frac {3 \cos {\left (x \right )}}{2} - \frac {\cos {\left (2 x \right )}}{2 \cos {\left (x \right )}} \]