Internal problem ID [7575]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 87.
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)]`]]
\[ \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 137
dsolve(x^2*(5+x+10*x^2)*diff(y(x),x$2)+x*(4+3*x+48*x^2)*diff(y(x),x)+(x+36*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{5}} \left (\frac {i \sqrt {199}+20 x +1}{i \sqrt {199}-20 x -1}\right )^{-\frac {i \sqrt {199}}{995}}}{10 x^{2}+x +5}+\frac {c_{2} x^{\frac {1}{5}} \left (\frac {i \sqrt {199}+20 x +1}{i \sqrt {199}-20 x -1}\right )^{-\frac {i \sqrt {199}}{995}} \left (\int \frac {\left (\frac {i \sqrt {199}-20 x -1}{i \sqrt {199}+20 x +1}\right )^{-\frac {i \sqrt {199}}{995}}}{x^{\frac {6}{5}}}d x \right )}{10 x^{2}+x +5} \]
✓ Solution by Mathematica
Time used: 2.147 (sec). Leaf size: 88
DSolve[x^2*(5+x+10*x^2)*y''[x]+x*(4+3*x+48*x^2)*y'[x]+(x+36*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt [5]{x} e^{-\frac {2 \arctan \left (\frac {20 x+1}{\sqrt {199}}\right )}{5 \sqrt {199}}} \left (c_2 \int _1^x\frac {e^{\frac {2 \arctan \left (\frac {20 K[1]+1}{\sqrt {199}}\right )}{5 \sqrt {199}}}}{K[1]^{6/5}}dK[1]+c_1\right )}{10 x^2+x+5} \]