Internal problem ID [8057]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 581.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 42
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)
\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+4\right )^{\frac {5}{8}}}{x^{\frac {1}{4}}}+\frac {c_{2} \left (x^{2}+4\right )^{\frac {5}{8}} \left (\int \frac {1}{\left (x^{2}+4\right )^{\frac {13}{8}} x}d x \right )}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.28 (sec). Leaf size: 198
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]
\[ y(x)\to \frac {c_2 \left (5\ 2^{3/4} \left (x^2+4\right )^{5/8} \arctan \left (\frac {\sqrt [8]{x^2+4}}{\sqrt [4]{2}}\right )+5 \sqrt [4]{2} \left (x^2+4\right )^{5/8} \arctan \left (\frac {\sqrt {2}-\sqrt [4]{x^2+4}}{2^{3/4} \sqrt [8]{x^2+4}}\right )-5\ 2^{3/4} \left (x^2+4\right )^{5/8} \text {arctanh}\left (\frac {\sqrt [8]{x^2+4}}{\sqrt [4]{2}}\right )+5 \sqrt [4]{2} \left (x^2+4\right )^{5/8} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt [8]{x^2+4}}{\sqrt {2} \sqrt [4]{x^2+4}+2}\right )+16\right )+80 c_1 \left (x^2+4\right )^{5/8}}{80 \sqrt [4]{x}} \]