Internal problem ID [8870]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1290.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (27 x^{2}+4\right ) y^{\prime \prime }+27 x y^{\prime }-3 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve((27*x^2+4)*diff(diff(y(x),x),x)+27*x*diff(y(x),x)-3*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sinh \left (\frac {\arcsinh \left (\frac {3 \sqrt {3}\, x}{2}\right )}{3}\right )+c_{2} \cosh \left (\frac {\arcsinh \left (\frac {3 \sqrt {3}\, x}{2}\right )}{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 103
DSolve[-3*y[x] + 27*x*y'[x] + (4 + 27*x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cosh \left (\frac {\sqrt {-27 x^2-4} \text {ArcTan}\left (\frac {3 x}{\sqrt {-9 x^2-\frac {4}{3}}}\right )}{3 \sqrt {27 x^2+4}}\right )+i c_2 \sinh \left (\frac {\sqrt {-27 x^2-4} \text {ArcTan}\left (\frac {3 x}{\sqrt {-9 x^2-\frac {4}{3}}}\right )}{3 \sqrt {27 x^2+4}}\right ) \\ \end{align*}