Internal problem ID [9000]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1421.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {a \left (-1+n \right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}+\frac {n \,a^{2} \left (\left (-1+n \right ) \left (\sin ^{2}\left (a x \right )\right )+\cos ^{2}\left (a x \right )\right ) y}{\cos \left (a x \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 27
dsolve(diff(diff(y(x),x),x) = -a*(n-1)*sin(2*a*x)/cos(a*x)^2*diff(y(x),x)-n*a^2*((n-1)*sin(a*x)^2+cos(a*x)^2)/cos(a*x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\cos ^{n}\left (a x \right )\right )+c_{2} \left (\cos ^{n -1}\left (a x \right )\right ) \sin \left (a x \right ) \]
✓ Solution by Mathematica
Time used: 0.11 (sec). Leaf size: 47
DSolve[y''[x] == -(a^2*n*Sec[a*x]^2*(Cos[a*x]^2 + (-1 + n)*Sin[a*x]^2)*y[x]) - a*(-1 + n)*Sec[a*x]^2*Sin[2*a*x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (2 a c_1-i c_2 e^{2 i a x}\right ) \cos ^n(a x)}{a \left (1+e^{2 i a x}\right )} \\ \end{align*}