Internal problem ID [7873]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 390.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x^{2} \left (x +2\right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 53
dsolve(2*x^2*(2+x)*diff(y(x), x$2) +5*x^2*diff(y(x),x)+(1+x)*y(x) = 0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (x +2\right )^{\frac {3}{2}}}+\frac {c_{2} \sqrt {2}\, \left (2 \sqrt {2}\, \sqrt {x +2}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right )\right ) \sqrt {x}}{2 \left (x +2\right )^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 55
DSolve[2*x^2*(2+x)*y''[x] +5*x^2*y'[x]+(1+x)*y[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt {x} \left (-2 \sqrt {2} c_2 \text {arctanh}\left (\frac {\sqrt {x+2}}{\sqrt {2}}\right )+2 c_2 \sqrt {x+2}+c_1\right )}{(x+2)^{3/2}} \]