Internal problem ID [9016]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1437.
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 {\left (3 \left (\sin ^{2}\relax (x )\right )+1\right ) y^{\prime }}{\cos \relax (x ) \sin \relax (x )}-\frac {y \left (\sin ^{2}\relax (x )\right )}{\cos \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.609 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x) = (3*sin(x)^2+1)/cos(x)/sin(x)*diff(y(x),x)+sin(x)^2/cos(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\cos ^{-\frac {3}{2}+\frac {\sqrt {13}}{2}}\relax (x )\right )+c_{2} \left (\cos ^{-\frac {3}{2}-\frac {\sqrt {13}}{2}}\relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.177 (sec). Leaf size: 36
DSolve[y''[x] == Tan[x]^2*y[x] + Csc[x]*Sec[x]*(1 + 3*Sin[x]^2)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \cos ^{-\frac {3}{2}-\frac {\sqrt {13}}{2}}(x) \left (c_2 \cos ^{\sqrt {13}}(x)+c_1\right ) \\ \end{align*}