Internal problem ID [10967]
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: 133.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 91
dsolve(x^2*diff(y(x),x$2)+a*x*diff(y(x),x)+x^n*(b*x^n+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {a}{2}-\frac {n}{2}+\frac {1}{2}} \operatorname {WhittakerM}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right )+c_{2} x^{-\frac {a}{2}-\frac {n}{2}+\frac {1}{2}} \operatorname {WhittakerW}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.269 (sec). Leaf size: 165
DSolve[x^2*y''[x]+a*x*y'[x]+x^n*(b*x^n+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {a+n-1}{2 n}} x^{\frac {1}{2} (-a-n+1)} \left (x^n\right )^{\frac {a+n-1}{2 n}} e^{\frac {i \sqrt {b} x^n}{n}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {-a+\frac {i c}{\sqrt {b}}-n+1}{2 n},\frac {a+n-1}{n},-\frac {2 i \sqrt {b} x^n}{n}\right )+c_2 L_{-\frac {a-\frac {i c}{\sqrt {b}}+n-1}{2 n}}^{\frac {a-1}{n}}\left (-\frac {2 i \sqrt {b} x^n}{n}\right )\right ) \]