Internal problem ID [10965]
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: 131.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 79
dsolve(x^2*diff(y(x),x$2)+lambda*x*diff(y(x),x)+(a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {\lambda }{2}} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}, 2 i \sqrt {a}\, x \right )+c_{2} x^{-\frac {\lambda }{2}} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {\lambda ^{2}-4 c -2 \lambda +1}}{2}, 2 i \sqrt {a}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.224 (sec). Leaf size: 159
DSolve[x^2*y''[x]+\[Lambda]*x*y'[x]+(a*x^2+b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-i \sqrt {a} x} x^{\frac {1}{2} \left (\sqrt {(\lambda -1)^2-4 c}-\lambda +1\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {(\lambda -1)^2-4 c}+1\right ),\sqrt {(\lambda -1)^2-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {(\lambda -1)^2-4 c}-1\right )}^{\sqrt {(\lambda -1)^2-4 c}}\left (2 i \sqrt {a} x\right )\right ) \]