Internal problem ID [9755]
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]]
\[ \boxed {y^{\prime \prime }+\frac {a \left (-1+n \right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}+\frac {n \,a^{2} \left (\left (-1+n \right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
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)
\[ y \left (x \right ) = c_{1} \sec \left (a x \right )^{-n +1} \sin \left (a x \right )+c_{2} \sec \left (a x \right )^{-n +1} \cos \left (a x \right ) \]
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 65
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]
\[ y(x)\to \frac {2^{-n} \left (2 a c_1-i c_2 e^{2 i a x}\right ) \left (e^{-i a x}+e^{i a x}\right )^n}{a \left (1+e^{2 i a x}\right )} \]