Internal problem ID [10943]
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: 119.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2 n} a^{2}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 134
dsolve(x^2*diff(y(x),x$2)-(a^2*x^(2*n)+a*(2*b+n-1)*x^n+b*(b-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = 2 \left (b -\frac {1}{2}-\frac {n}{2}\right )^{2} c_{2} x^{-\frac {3 n}{2}+\frac {1}{2}} \operatorname {WhittakerM}\left (\frac {n -2 b +1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+n \left (\left (-b +\frac {1}{2}+\frac {n}{2}\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}+x^{-\frac {n}{2}+\frac {1}{2}} a \right ) c_{2} \operatorname {WhittakerM}\left (-\frac {2 b +n -1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+c_{1} x^{b} {\mathrm e}^{\frac {a \,x^{n}}{n}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*y''[x]-(a^2*x^(2*n)+a*(2*b+n-1)*x^n+b*(b-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved