Internal problem ID [10869]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 35.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+2 b \right ) y^{\prime }+\left (b a \,x^{2}-a x +b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 74
dsolve(diff(y(x),x$2)+(a*x^2+2*b)*diff(y(x),x)+(a*b*x^2-a*x+b^2)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \,{\mathrm e}^{-b x}+\frac {c_{2} {\mathrm e}^{-\frac {x \left (a \,x^{2}+6 b \right )}{6}} \left (x^{6} a^{2} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {a \,x^{3}}{3}\right )+5 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {a \,x^{3}}{3}\right ) a \,x^{3}+10 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {a \,x^{3}}{3}\right )\right )}{x^{4}} \]
✓ Solution by Mathematica
Time used: 0.407 (sec). Leaf size: 51
DSolve[y''[x]+(a*x^2+2*b)*y'[x]+(a*b*x^2-a*x+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{9} e^{-b x} \left (9 c_1 x-3^{2/3} c_2 \sqrt [3]{a x^3} \Gamma \left (-\frac {1}{3},\frac {a x^3}{3}\right )\right ) \]