Internal problem ID [9097]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1518.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 56
dsolve((x^2+1)*x*diff(diff(diff(y(x),x),x),x)+3*(2*x^2+1)*diff(diff(y(x),x),x)-12*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \sqrt {x^{2}+1}+\frac {c_{2} \left (3 x^{2} \sqrt {x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )-3 x^{2}-1\right )}{x}+c_{3} \left (2 x^{2}+1\right ) \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 69
DSolve[-12*y[x] + 3*(1 + 2*x^2)*y''[x] + x*(1 + x^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (c_1 \left (4 x^2+2\right )+2 c_2 x \sqrt {x^2+1}-3 c_3 x \sqrt {x^2+1} \tanh ^{-1}\left (\sqrt {x^2+1}\right )+3 c_3 x+\frac {c_3}{x}\right ) \\ \end{align*}