58.2.25 problem 25

Internal problem ID [9148]
Book : Second order enumerated odes
Section : section 2
Problem number : 25
Date solved : Sunday, March 30, 2025 at 02:23:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \cos \left (x \right ) y^{\prime \prime }+\sin \left (x \right ) y^{\prime }-2 y \cos \left (x \right )^{3}&=2 \cos \left (x \right )^{5} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 30

ode:=cos(x)*diff(diff(y(x),x),x)+sin(x)*diff(y(x),x)-2*y(x)*cos(x)^3 = 2*cos(x)^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sinh \left (\sin \left (x \right ) \sqrt {2}\right ) c_2 +\cosh \left (\sin \left (x \right ) \sqrt {2}\right ) c_1 +\frac {1}{2}-\frac {\cos \left (2 x \right )}{2} \]

✓ Mathematica. Time used: 11.572 (sec). Leaf size: 90

ode=Cos[x]*D[y[x],{x,2}]+Sin[x]*D[y[x],x]-2*y[x]*Cos[x]^3==2*Cos[x]^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-i \sqrt {\cos (2 x)-1}} \left (\cos \left (\sqrt {\cos (2 x)-1}\right )+i \sin \left (\sqrt {\cos (2 x)-1}\right )\right ) \left (-\cos (2 x)+2 c_1 \cos \left (\sqrt {\cos (2 x)-1}\right )+2 c_2 \sin \left (\sqrt {\cos (2 x)-1}\right )+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x)*cos(x)**3 + sin(x)*Derivative(y(x), x) - 2*cos(x)**5 + cos(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(2*y(x)*cos(x)**2 + 2*cos(x)**4 - Derivative(y(x), (x, 2)))/tan(x) + Derivative(y(x), x) cannot be solved by the factorable group method