2.1.66 Problem 66

Internal problem ID [10052]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 66
Date solved : Thursday, November 27, 2025 at 10:07:36 AM
CAS classification : [NONE]

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

✗ Maple

ode:=3*y(x)*diff(diff(y(x),x),x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one \ 
integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the sing\ 
ular cases 
trying symmetries linear in x and y(x) 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(y) 
trying 2nd order, integrating factor of the form mu(x,y) 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the gener\ 
al case 
trying 2nd order, integrating factor of the form mu(y,y) 
trying differential order: 2; mu polynomial in y 
trying 2nd order, integrating factor of the form mu(x,y) 
differential order: 2; looking for linear symmetries 
-> trying 2nd order, the S-function method 
   -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering metho\ 
ds for the S-function 
   -> trying 2nd order, the S-function method 
   -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the gener\ 
al case 
-> trying 2nd order, dynamical_symmetries, only a reduction of order through on\ 
e integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
--- Trying Lie symmetry methods, 2nd order --- 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 5 
   -> Computing symmetries using: way = formal
 

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : solve: Cannot solve 3*y(x)*Derivative(y(x), (x, 2)) - sin(x)