23.3.37 problem 37

Internal problem ID [5751]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 37
Date solved : Friday, October 03, 2025 at 01:43:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 47

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

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

✗ Sympy

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

False