Internal problem ID [10321]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } \left (\sin ^{2}\relax (x )\right )-2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 57
dsolve(sin(x)^2*diff(y(x),x$2)-2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sin \left (2 x \right )}{-1+\cos \left (2 x \right )}+\frac {c_{2} \left (-i \sin \left (2 x \right ) \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right )+2 \cos \left (2 x \right )-2\right )}{-1+\cos \left (2 x \right )} \]
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 46
DSolve[Sin[x]^2*y''[x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\cos (x) \left (c_1-c_2 \log \left (\sqrt {-\sin ^2(x)}-\cos (x)\right )\right )}{\sqrt {-\sin ^2(x)}}-c_2 \\ \end{align*}