Internal problem ID [8927]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1348.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 101
dsolve(diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-(b*x^2+a*(x^4+1))/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \mathit {HD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )+c_{2} \mathit {HD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right ) \left (\int \frac {1}{x \mathit {HD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 34
DSolve[y''[x] == -(((b*x^2 + a*(1 + x^4))*y[x])/x^4) - y'[x]/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {MathieuC}[-b,a,i \log (x)]+c_2 \text {MathieuS}[-b,a,i \log (x)] \\ \end{align*}