23.3.194 problem 196

Internal problem ID [5908]
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 : 196
Date solved : Friday, October 03, 2025 at 01:45:18 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} -y+\left (1+a \right ) y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=-y(x)+(a+1)*diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_2 +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right ) c_1 \right ) x^{-\frac {a}{2}} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 58

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

\begin{align*} y(x)&\to (-1)^{-a} x^{-a/2} \left (c_2 \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (a + 1)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: -a