23.3.331 problem 333

Internal problem ID [6045]
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 : 333
Date solved : Tuesday, September 30, 2025 at 02:20:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 a x +5\right ) y-x \left (a x +5\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

ode:=(3*a*x+5)*y(x)-x*(a*x+5)*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_1 \left (a^{2} x^{2}+4 a x +6\right )+c_2 \,{\mathrm e}^{a x} \left (a x -3\right )\right ) \]

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 44

ode=(5 + 3*a*x)*y[x] - x*(5 + a*x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x \left (a^2 c_1 e^{a x} (a x-3)-c_2 \left (a^2 x^2+4 a x+6\right )\right )}{a^3} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(a*x + 5)*Derivative(y(x), x) + (3*a*x + 5)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None