23.3.328 problem 330

Internal problem ID [6042]
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 : 330
Date solved : Friday, October 03, 2025 at 01:45:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 93

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

\[ y = -\frac {{\mathrm e}^{-\frac {x}{2}} \left (-c_1 \left (\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )-c_1 \left (-\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\left (\left (-\sqrt {2}-x -1\right ) \operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )\right ) c_2 \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 63

ode=-y[x] + x*(3 + 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 e^{-x} x^{\sqrt {2}-1} \left (c_1 \operatorname {HypergeometricU}\left (2+\sqrt {2},1+2 \sqrt {2},x\right )+c_2 L_{-2-\sqrt {2}}^{2 \sqrt {2}}(x)\right ) \end{align*}

✓ Sympy. Time used: 0.841 (sec). Leaf size: 473

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

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