23.3.279 problem 281

Internal problem ID [5993]
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 : 281
Date solved : Friday, October 03, 2025 at 01:45:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (m +1\right ) x^{m} a \left (m \right ) y+x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 49

ode:=(1+m)*x^m*a(m)*y(x)+x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {BesselJ}\left (0, \frac {2 \sqrt {\left (m +1\right ) a \left (m \right )}\, x^{\frac {m}{2}}}{m}\right )+c_2 \operatorname {BesselY}\left (0, \frac {2 \sqrt {\left (m +1\right ) a \left (m \right )}\, x^{\frac {m}{2}}}{m}\right ) \]

✓ Mathematica. Time used: 0.963 (sec). Leaf size: 63

ode=(1 + m)*x^m*a[m]*y[x] + 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 c_1 \operatorname {BesselI}\left (0,\frac {2 \sqrt {x^m} \sqrt {-((m+1) a(m))}}{m}\right )+2 c_2 K_0\left (\frac {2 \sqrt {x^m} \sqrt {-((m+1) a(m))}}{m}\right ) \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{1} J_{0}\left (\frac {2 x^{\frac {m}{2}} \sqrt {\left (m + 1\right ) a{\left (m \right )}}}{m}\right ) + C_{2} Y_{0}\left (\frac {2 x^{\frac {m}{2}} \sqrt {\left (m + 1\right ) a{\left (m \right )}}}{m}\right ) \]