Internal problem ID [10280]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y^{\prime } \tan \relax (x )-\left (a^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve(diff(y(x),x$2)-2*tan(x)*diff(y(x),x)-(a^2+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sinh \left (a x \right )}{\cos \relax (x )}+\frac {c_{2} \cosh \left (a x \right )}{\cos \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 32
DSolve[y''[x]-2*Tan[x]*y'[x]-(a^2+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sec (x) \left (c_1 e^{-a x}+\frac {c_2 e^{a x}}{2 a}\right ) \\ \end{align*}