Internal problem ID [10843]
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 Equations Containing
Power Functions. page 213
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-a \,x^{n -2} \left (x^{n} a +n +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.562 (sec). Leaf size: 109
dsolve(diff(y(x),x$2)-a*x^(n-2)*(a*x^n+n+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{\frac {a \,x^{n}}{n}}+c_{2} \left (\left (\left (n^{2}-n \right ) x^{-\frac {3 n}{2}+\frac {1}{2}}+2 a n \,x^{-\frac {n}{2}+\frac {1}{2}}\right ) \operatorname {WhittakerM}\left (-\frac {1}{2}-\frac {1}{2 n}, -\frac {1}{2 n}+1, \frac {2 a \,x^{n}}{n}\right )+x^{-\frac {3 n}{2}+\frac {1}{2}} \operatorname {WhittakerM}\left (\frac {1}{2}-\frac {1}{2 n}, -\frac {1}{2 n}+1, \frac {2 a \,x^{n}}{n}\right ) \left (n -1\right )^{2}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]-a*x^(n-2)*(a*x^n+n+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved