Internal problem ID [10910]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 86.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 107
dsolve(x*diff(y(x),x$2)+(a*x^3+b)*diff(y(x),x)+a*(b-1)*x^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {9 c_{2} a^{2} x^{-\frac {b}{2}+3} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )+\left (a \,x^{-\frac {b}{2}+3}+x^{-\frac {b}{2}} \left (b +2\right )\right ) c_{2} {\mathrm e}^{-\frac {a \,x^{3}}{3}} a 3^{-\frac {b}{6}+\frac {2}{3}} \left (b +5\right ) \left (a \,x^{3}\right )^{\frac {1}{3}+\frac {b}{6}}+9 c_{1} x^{-b +2}}{9 x} \]
✓ Solution by Mathematica
Time used: 0.424 (sec). Leaf size: 60
DSolve[x*y''[x]+(a*x^3+b)*y'[x]+a*(b-1)*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 x^{1-b}-3^{\frac {b-4}{3}} c_2 \left (a x^3\right )^{\frac {1}{3}-\frac {b}{3}} \Gamma \left (\frac {b-1}{3},\frac {a x^3}{3}\right ) \]