23.3.309 problem 311

Internal problem ID [6023]
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 : 311
Date solved : Tuesday, September 30, 2025 at 02:20:03 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \operatorname {a2} y+\operatorname {a1} x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

ode:=a2*y(x)+a1*x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {\operatorname {a1}}{2}} \sqrt {x}\, \left (x^{\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}} c_1 +x^{-\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}} c_2 \right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 57

ode=a2*y[x] + a1*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 x^{\frac {1}{2} \left (-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} \left (c_2 x^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.191 (sec). Leaf size: 617

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
y = Function("y") 
ode = Eq(a1*x*Derivative(y(x), x) + a2*y(x) + x**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]