62.37.4 problem Ex 4

Internal problem ID [13011]
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 : Tuesday, January 28, 2025 at 08:24:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.755 (sec). Leaf size: 29

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

\[ y = -i \cot \left (x \right ) \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) c_{2} +c_{1} \cot \left (x \right )-2 c_{2} \]

✓ Solution by Mathematica

Time used: 0.202 (sec). Leaf size: 42

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

\[ y(x)\to -\frac {c_2 \cos (x) \arcsin (\cos (x))}{\sqrt {\sin ^2(x)}}+\frac {c_1 \cos (x)}{\sqrt {-\sin ^2(x)}}-c_2 \]