Internal problem ID [8922]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1343.
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 {\left (x^{2} a \left (-a +1\right )-b \left (x +b \right )\right ) y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 62
dsolve(diff(diff(y(x),x),x) = -(x^2*a*(1-a)-b*(x+b))/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\left (2 a x +b \right ) \BesselI \left (a , \frac {b}{x}\right )+\BesselI \left (a +1, \frac {b}{x}\right ) b \right )+c_{2} \left (\left (2 a x +b \right ) \BesselK \left (a , \frac {b}{x}\right )-\BesselK \left (a +1, \frac {b}{x}\right ) b \right ) \]
✓ Solution by Mathematica
Time used: 0.128 (sec). Leaf size: 65
DSolve[y''[x] == -((((1 - a)*a*x^2 - b*(b + x))*y[x])/x^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 (2 a x+b) I_a\left (\frac {b}{x}\right )+b c_1 I_{a+1}\left (\frac {b}{x}\right )+c_2 \left ((2 a x+b) K_a\left (\frac {b}{x}\right )-b K_{a+1}\left (\frac {b}{x}\right )\right ) \\ \end{align*}