Internal problem ID [7299]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 579.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.417 (sec). Leaf size: 85
dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)+7*x^3*diff(y(x),x)+(1+3*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sqrt {x}}{\left (x^{2}+2\right )^{\frac {3}{4}}}+\frac {c_{2} \sqrt {x}\, \left (-\ln \left (1-\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )+\ln \left (1+\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )-2 \arctan \left (\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )\right )}{\left (2 x^{2}+4\right )^{\frac {3}{4}}} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 71
DSolve[2*x^2*(2+x^2)*y''[x]+7*x^3*y'[x]+(1+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {x} \left (2^{3/4} c_2 \left (\operatorname {ArcTan}\left (\frac {\sqrt [4]{x^2+2}}{\sqrt [4]{2}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{x^2+2}}{\sqrt [4]{2}}\right )\right )+2 c_1\right )}{2 \left (x^2+2\right )^{3/4}} \\ \end{align*}