62.30.6 problem Ex 6

Internal problem ID [12970]
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 53. Change of dependent variable. Page 125
Problem number : Ex 6
Date solved : Tuesday, January 28, 2025 at 04:45:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 20

dsolve(diff(y(x),x$2)-2*tan(x)*diff(y(x),x)-(a^2+1)*y(x)=0,y(x), singsol=all)
 

\[ y = \sec \left (x \right ) \left (c_{1} \sinh \left (a x \right )+c_{2} \cosh \left (a x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 32

DSolve[D[y[x],{x,2}]-2*Tan[x]*D[y[x],x]-(a^2+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sec (x) \left (c_1 e^{-a x}+\frac {c_2 e^{a x}}{2 a}\right ) \]