69.20.6 problem 641

Internal problem ID [18427]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 641
Date solved : Thursday, October 02, 2025 at 03:11:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\cos \left (\sin \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x)+tan(x)*diff(y(x),x)+cos(x)^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (\sin \left (x \right )\right )+c_2 \cos \left (\sin \left (x \right )\right ) \]

✓ Mathematica. Time used: 1.201 (sec). Leaf size: 37

ode=D[y[x],{x,2}]+Tan[x]*D[y[x],x]+Cos[x]^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cosh \left (\sqrt {-\sin ^2(x)}\right )+i c_2 \sinh \left (\sqrt {-\sin ^2(x)}\right ) \end{align*}

✗ Sympy

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

False