Internal problem ID [10979]
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: 145.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x^{n} a +b \right ) y^{\prime }+b \left (x^{n} a -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 132
dsolve(x^2*diff(y(x),x$2)+x*(a*x^n+b)*diff(y(x),x)+b*(a*x^n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-b}+c_{2} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (n \left (\left (b +n +1\right ) x^{\frac {1}{2}-\frac {3 n}{2}-\frac {b}{2}}+a \,x^{\frac {1}{2}-\frac {b}{2}-\frac {n}{2}}\right ) \operatorname {WhittakerM}\left (\frac {1+b -n}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right )+x^{\frac {1}{2}-\frac {3 n}{2}-\frac {b}{2}} \operatorname {WhittakerM}\left (\frac {b +n +1}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right ) \left (b +n +1\right )^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 76
DSolve[x^2*y''[x]+x*(a*x^n+b)*y'[x]+b*(a*x^n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-\frac {b}{n}} n^{\frac {b}{n}-1} a^{-\frac {b}{n}} \left (x^n\right )^{-\frac {b}{n}} \left ((b+1) c_1 (-1)^{b/n} \Gamma \left (\frac {b+1}{n},0,\frac {a x^n}{n}\right )+c_2 n\right ) \]