88.21.6 problem 6

Internal problem ID [24156]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 149
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:00:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (3 x -9\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-3 x} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+x*diff(y(x),x)+(3*x-9)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x -6\right )}{2}\right ) c_1 +c_2 \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 45

ode=D[y[x],{x,2}]+x*D[y[x],{x,1}]+(3*x-9)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {1}{2} (x-6) x} \left (c_1 \operatorname {HermiteH}\left (-1,\frac {x-6}{\sqrt {2}}\right )+c_2 e^{\frac {1}{2} (x-6)^2}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + (3*x - 9)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False