23.3.164 problem 166

Internal problem ID [5878]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 166
Date solved : Tuesday, September 30, 2025 at 02:05:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} -\left (a^{2}+1\right ) y-2 \tan \left (x \right ) y^{\prime }+y^{\prime \prime }&=\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 32

ode:=-(a^2+1)*y(x)-2*tan(x)*diff(y(x),x)+diff(diff(y(x),x),x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sec \left (x \right ) \sinh \left (a x \right ) c_2 +\sec \left (x \right ) \cosh \left (a x \right ) c_1 -\frac {\sin \left (x \right )}{a^{2}+4} \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 46

ode=-((1 + a^2)*y[x]) - 2*Tan[x]*D[y[x],x] + D[y[x],{x,2}] == Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \sec (x) \left (-\frac {\sin (2 x)}{a^2+4}+2 c_1 e^{-a x}+\frac {c_2 e^{a x}}{a}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(-a**2*y(x) - y(x) - sin(x) + Derivative(y(x), (x, 2)))/(2*tan(