Internal problem ID [6879]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 147.
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}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.215 (sec). Leaf size: 68
dsolve(4*x^2*(4+x^2)*diff(y(x),x$2)+3*x*(8+3*x^2)*diff(y(x),x)+(1-9*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (x^{2}+4\right )^{\frac {5}{8}}}{x^{\frac {1}{4}}}+\frac {c_{2} \left (-\frac {1024}{25}+\left (x^{2} \hypergeom \left (\left [1, 1, \frac {13}{8}\right ], \left [2, 2\right ], -\frac {x^{2}}{4}\right )-\frac {32 \mcoloneq \gamma }{5}+\frac {64 \ln \relax (2)}{5}-\frac {64 \ln \relax (x )}{5}-\frac {32 \Psi \left (\frac {5}{8}\right )}{5}\right ) \left (x^{2}+4\right )^{\frac {5}{8}} 2^{\frac {3}{4}}\right )}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.267 (sec). Leaf size: 185
DSolve[4*x^2*(4+x^2)*y''[x]+3*x*(8+3*x^2)*y'[x]+(1-9*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {5 \sqrt [4]{2} c_2 \left (x^2+4\right )^{5/8} \left (\sqrt {2} \operatorname {ArcTan}\left (\frac {\sqrt [8]{x^2+4}}{\sqrt [4]{2}}\right )+\operatorname {ArcTan}\left (\frac {\sqrt {2}-\sqrt [4]{x^2+4}}{2^{3/4} \sqrt [8]{x^2+4}}\right )\right )+16 \left (5 c_1 \left (x^2+4\right )^{5/8}+c_2\right )+5 \sqrt [4]{2} c_2 \left (x^2+4\right )^{5/8} \left (\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [8]{x^2+4}}{\sqrt {2} \sqrt [4]{x^2+4}+2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [8]{x^2+4}}{\sqrt [4]{2}}\right )\right )}{80 \sqrt [4]{x}} \\ \end{align*}