61.27.25 problem 35

Internal problem ID [12535]
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
Problem number : 35
Date solved : Tuesday, January 28, 2025 at 03:20:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 90

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)
 

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

✓ Solution by Mathematica

Time used: 0.363 (sec). Leaf size: 50

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

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