56.37.4 problem Ex 4

Internal problem ID [14292]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 61. Transformation of variables. Page 143
Problem number : Ex 4
Date solved : Friday, October 03, 2025 at 07:29:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right )^{2} y^{\prime \prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 29

ode:=sin(x)^2*diff(diff(y(x),x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -i \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) c_2 \cot \left (x \right )+c_1 \cot \left (x \right )-2 c_2 \]

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to \frac {c_2 \cos (x) \text {arctanh}\left (\frac {\cos (x)}{\sqrt {-\sin ^2(x)}}\right )+c_1 \cos (x)-c_2 \sqrt {-\sin ^2(x)}}{\sqrt {-\sin ^2(x)}} \end{align*}

✗ Sympy

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

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