77.1.107 problem 135 (page 195)

Internal problem ID [17997]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 135 (page 195)
Date solved : Tuesday, January 28, 2025 at 08:28:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.168 (sec). Leaf size: 29

dsolve(diff(y(x),x$2)*sin(x)^2=2*y(x),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: 0.146 (sec). Leaf size: 55

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

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