Internal problem ID [10969]
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: 135.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+x c +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 83
dsolve(x^2*diff(y(x),x$2)+a*x^2*diff(y(x),x)+(b*x^2+c*x+d)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a x}{2}} \operatorname {WhittakerM}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right )+c_{2} {\mathrm e}^{-\frac {a x}{2}} \operatorname {WhittakerW}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.228 (sec). Leaf size: 157
DSolve[x^2*y''[x]+a*x^2*y'[x]+(b*x^2+c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{\frac {1}{2} \left (\sqrt {1-4 d}+1\right )} e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (-\frac {2 c}{\sqrt {a^2-4 b}}+\sqrt {1-4 d}+1\right ),\sqrt {1-4 d}+1,\sqrt {a^2-4 b} x\right )+c_2 L_{\frac {c}{\sqrt {a^2-4 b}}-\frac {1}{2} \sqrt {1-4 d}-\frac {1}{2}}^{\sqrt {1-4 d}}\left (\sqrt {a^2-4 b} x\right )\right ) \]