61.26.4 problem 4

Internal problem ID [12425]
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 : 4
Date solved : Monday, March 31, 2025 at 05:34:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 43

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

\[ y = \frac {c_1 \operatorname {WhittakerM}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )+c_2 \operatorname {WhittakerW}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 68

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

\[ y(x)\to c_1 \operatorname {ParabolicCylinderD}\left (-\frac {b}{2 \sqrt {a}}-\frac {1}{2},\sqrt {2} \sqrt [4]{a} x\right )+c_2 \operatorname {ParabolicCylinderD}\left (\frac {1}{2} \left (\frac {b}{\sqrt {a}}-1\right ),i \sqrt {2} \sqrt [4]{a} x\right ) \]

✗ Sympy

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

False