61.28.4 problem 64

Internal problem ID [12564]
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 : 64
Date solved : Tuesday, January 28, 2025 at 08:02:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.171 (sec). Leaf size: 61

dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+(b*x+c)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-i \sqrt {b}\, x} \left (\operatorname {KummerM}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.058 (sec). Leaf size: 85

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

\[ y(x)\to e^{-i \sqrt {b} x} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (a+\frac {i c}{\sqrt {b}}\right ),a,2 i \sqrt {b} x\right )+c_2 L_{-\frac {a}{2}-\frac {i c}{2 \sqrt {b}}}^{a-1}\left (2 i \sqrt {b} x\right )\right ) \]