Internal problem ID [10844]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{-1+n}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.86 (sec). Leaf size: 95
dsolve(diff(y(x),x$2)+(a*x^(2*n)+b*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{\sqrt {a}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, x^{1+n}}{1+n}\right ) x^{-\frac {n}{2}}+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{\sqrt {a}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, x^{1+n}}{1+n}\right ) x^{-\frac {n}{2}} \]
✓ Solution by Mathematica
Time used: 0.399 (sec). Leaf size: 225
DSolve[y''[x]+(a*x^(2*n)+b*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} e^{-\frac {\sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}},\frac {n}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_2 L_{\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right ) \]