Internal problem ID [7268]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 548.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 42
dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+4*x*(1+6*x^2)*diff(y(x),x)-(1-25*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sqrt {x}}{\left (x^{2}+1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (-\arcsinh \relax (x ) x +\sqrt {x^{2}+1}\right )}{\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 54
DSolve[4*x^2*(1+x^2)*y''[x]+4*x*(1+6*x^2)*y'[x]-(1-25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-c_2 \sqrt {x^2+1}+c_2 x \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )+c_1 x}{\sqrt {x} \left (x^2+1\right )^{3/2}} \\ \end{align*}