61.28.26 problem 86

Internal problem ID [12586]
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
Problem number : 86
Date solved : Tuesday, January 28, 2025 at 03:22:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (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)
 

\[ y = \frac {9 c_{2} a^{2} x^{-\frac {b}{2}+3} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )+\left (a \,x^{-\frac {b}{2}+3}+x^{-\frac {b}{2}} \left (b +2\right )\right ) a c_{2} 3^{-\frac {b}{6}+\frac {2}{3}} \left (b +5\right ) {\mathrm e}^{-\frac {a \,x^{3}}{3}} \left (a \,x^{3}\right )^{\frac {1}{3}+\frac {b}{6}}+9 c_{1} x^{-b +2}}{9 x} \]

✓ Solution by Mathematica

Time used: 0.474 (sec). Leaf size: 51

DSolve[x*D[y[x],{x,2}]+(a*x^3+b)*D[y[x],x]+a*(b-1)*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^{1-b} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {-a K[1]^3+b-2}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]