55.27.6 problem 6

Internal problem ID [13779]
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 Equations Containing Power Functions. page 213
Problem number : 6
Date solved : Friday, October 03, 2025 at 06:54:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 142

ode:=diff(diff(y(x),x),x)-(a*x^2+b*c*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x \left (a x +b c \right )}{2 \sqrt {a}}} \left (2 \operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-12 a^{{3}/{2}}}{16 a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{{3}/{2}}}\right ) c_2 a x +\operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-12 a^{{3}/{2}}}{16 a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{{3}/{2}}}\right ) c_2 b c +c_1 \operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-4 a^{{3}/{2}}}{16 a^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{{3}/{2}}}\right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 92

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

\begin{align*} y(x)&\to c_2 \operatorname {ParabolicCylinderD}\left (-\frac {b^2 c^2}{8 a^{3/2}}-\frac {1}{2},\frac {i (b c+2 a x)}{\sqrt {2} a^{3/4}}\right )+c_1 \operatorname {ParabolicCylinderD}\left (\frac {1}{8} \left (\frac {b^2 c^2}{a^{3/2}}-4\right ),\frac {b c+2 a x}{\sqrt {2} a^{3/4}}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((-a*x**2 - b*c*x)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False