23.3.305 problem 307

Internal problem ID [6019]
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 : 307
Date solved : Tuesday, September 30, 2025 at 02:19:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+6\right ) y+4 x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 29

ode:=(x^2+6)*y(x)+4*x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \operatorname {BesselJ}\left (\frac {i \sqrt {15}}{2}, x\right )+c_2 \operatorname {BesselY}\left (\frac {i \sqrt {15}}{2}, x\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 44

ode=(6 + x^2)*y[x] + 4*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 {c_1 \operatorname {BesselJ}\left (\frac {i \sqrt {15}}{2},x\right )+c_2 \operatorname {BesselY}\left (\frac {i \sqrt {15}}{2},x\right )}{x^{3/2}} \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (x**2 + 6)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {\sqrt {15} i}{2}}\left (x\right ) + C_{2} Y_{\frac {\sqrt {15} i}{2}}\left (x\right )}{x^{\frac {3}{2}}} \]