Internal problem ID [10954]
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: 120.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 95
dsolve(x^2*diff(y(x),x$2)+(a*x^(2*n)+b*x^n+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {n}{2}+\frac {1}{2}} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}\, n}, \frac {i \sqrt {-1+4 c}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right )+c_{2} x^{-\frac {n}{2}+\frac {1}{2}} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}\, n}, \frac {i \sqrt {-1+4 c}}{2 n}, \frac {2 i \sqrt {a}\, x^{n}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.313 (sec). Leaf size: 236
DSolve[x^2*y''[x]+(a*x^(2*n)+b*x^n+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {\sqrt {(1-4 c) n^2}+n^2}{2 n^2}} x^{\frac {1}{2}-\frac {n}{2}} e^{\frac {i \sqrt {a} x^n}{n}} \left (x^n\right )^{\frac {\sqrt {(1-4 c) n^2}+n^2}{2 n^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (-\frac {i b}{\sqrt {a} n}+\frac {\sqrt {(1-4 c) n^2}}{n^2}+1\right ),\frac {\sqrt {(1-4 c) n^2}}{n^2}+1,-\frac {2 i \sqrt {a} x^n}{n}\right )+c_2 L_{\frac {1}{2} \left (\frac {i b}{\sqrt {a} n}-\frac {\sqrt {(1-4 c) n^2}}{n^2}-1\right )}^{\frac {\sqrt {(1-4 c) n^2}}{n^2}}\left (-\frac {2 i \sqrt {a} x^n}{n}\right )\right ) \]