Internal problem ID [10941]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 117.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (-1+2 b \right ) x^{2}+b \left (1+b \right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve(x^2*diff(y(x),x$2)-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-b} {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_{2} \Gamma \left (b +\frac {1}{2}\right )-c_{2} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.45 (sec). Leaf size: 66
DSolve[x^2*y''[x]-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-\frac {a x^2}{2}} x^{-b} \left (a c_2 x^{2 b+3} \left (-a x^2\right )^{-b-\frac {3}{2}} \Gamma \left (b+\frac {1}{2},-a x^2\right )+2 c_1\right ) \]