67.9.2 problem 14.1 (b)

Internal problem ID [16550]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.1 (b)
Date solved : Friday, October 03, 2025 at 07:30:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 54

ode:=diff(diff(y(x),x),x)+x^2*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right ) {\mathrm e}^{-\frac {x^{3}}{3}} \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{{2}/{3}}, -3, 0, \frac {3^{{2}/{3}} x}{3}\right )^{2}}d x c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 35

ode=D[y[x],{x,2}]+x^2*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 e^{-\frac {x^3}{3}} \text {HeunT}[4,-2,0,0,-1,x]+c_1 \text {HeunT}[4,0,0,0,1,x] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) - 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False