Internal problem ID [7136]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 412.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 75
dsolve((x^2+3)*diff(y(x),x$2)-7*x*diff(y(x),x)+16*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{4}-9 x^{2}+\frac {27}{8}\right )+c_{2} \left (\frac {\left (96 x^{4}-864 x^{2}+324\right ) \ln \left (\sqrt {x^{2}+3}-x \right )}{6144}+\frac {\left (200 x^{3}-660 x \right ) \sqrt {x^{2}+3}}{6144}+\frac {25 x^{4}}{768}-\frac {75 x^{2}}{256}+\frac {225}{2048}\right ) \]
✓ Solution by Mathematica
Time used: 0.186 (sec). Leaf size: 492
DSolve[(x^2+3)*y''[x]-7*x*y'[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{24} c_2 \left (12960 x^2 \text {RootSum}\left [7838208000 \text {$\#$1}^4-188281584000 \text {$\#$1}^2-241544908800 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (-411757211968704000 \text {$\#$1}^3-166063274606980800 \text {$\#$1}^2+10138703825167113960 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]+5248800 x^2 \text {RootSum}\left [210880720572480000000 \text {$\#$1}^4-30882886815600000 \text {$\#$1}^2+97825688064000 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (27353083060732502808000000 \text {$\#$1}^3-27238528617410025720000 \text {$\#$1}^2-4106175049192681153800 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]-4860 \text {RootSum}\left [7838208000 \text {$\#$1}^4-188281584000 \text {$\#$1}^2-241544908800 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (-411757211968704000 \text {$\#$1}^3-166063274606980800 \text {$\#$1}^2+10138703825167113960 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]-1968300 \text {RootSum}\left [210880720572480000000 \text {$\#$1}^4-30882886815600000 \text {$\#$1}^2+97825688064000 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (27353083060732502808000000 \text {$\#$1}^3-27238528617410025720000 \text {$\#$1}^2-4106175049192681153800 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]-1440 x^4 \text {RootSum}\left [7838208000 \text {$\#$1}^4-188281584000 \text {$\#$1}^2-241544908800 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (-411757211968704000 \text {$\#$1}^3-166063274606980800 \text {$\#$1}^2+10138703825167113960 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]-583200 x^4 \text {RootSum}\left [210880720572480000000 \text {$\#$1}^4-30882886815600000 \text {$\#$1}^2+97825688064000 \text {$\#$1}+18453344881\&,\text {$\#$1} \log \left (27353083060732502808000000 \text {$\#$1}^3-27238528617410025720000 \text {$\#$1}^2-4106175049192681153800 \text {$\#$1}-868082003147887664 x^2+868082003147887664 x \sqrt {x^2+3}+15417510572689690113\right )\&\right ]+165 \sqrt {x^2+3} x+216 x^2 \log \left (\sqrt {x^2+3}-x\right )-81 \log \left (\sqrt {x^2+3}-x\right )-24 x^4 \log \left (\sqrt {x^2+3}-x\right )-50 \sqrt {x^2+3} x^3\right )+c_1 \left (x^4-9 x^2+\frac {27}{8}\right ) \\ \end{align*}