62.31.3 problem Ex 3

Internal problem ID [12975]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 3
Date solved : Tuesday, January 28, 2025 at 04:46:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.247 (sec). Leaf size: 15

dsolve(diff(y(x),x$2)+tan(x)*diff(y(x),x)+cos(x)^2*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.183 (sec). Leaf size: 37

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

\[ y(x)\to c_1 \cosh \left (\sqrt {-\sin ^2(x)}\right )+i c_2 \sinh \left (\sqrt {-\sin ^2(x)}\right ) \]