60.3.415 problem 1421

Internal problem ID [11425]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1421
Date solved : Monday, January 27, 2025 at 11:20:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {a \left (n -1\right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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)
 

\[ y = \sec \left (a x \right )^{-n +1} \left (c_{1} \sin \left (a x \right )+c_{2} \cos \left (a x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.190 (sec). Leaf size: 65

DSolve[D[y[x],{x,2}] == -(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]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2^{-n} \left (2 a c_1-i c_2 e^{2 i a x}\right ) \left (e^{-i a x}+e^{i a x}\right )^n}{a \left (1+e^{2 i a x}\right )} \]