23.3.233 problem 235

Internal problem ID [5947]
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 : 235
Date solved : Friday, October 03, 2025 at 01:45:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 84

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

\[ y = \sqrt {x}\, {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {b}\, x}{2}} \left (\operatorname {KummerU}\left (\frac {i a \sqrt {2}+3 \sqrt {b}}{4 \sqrt {b}}, \frac {3}{2}, i \sqrt {2}\, \sqrt {b}\, x \right ) c_2 +\operatorname {KummerM}\left (\frac {i a \sqrt {2}+3 \sqrt {b}}{4 \sqrt {b}}, \frac {3}{2}, i \sqrt {2}\, \sqrt {b}\, x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 115

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

\begin{align*} y(x)&\to \sqrt {x} e^{-\frac {i \sqrt {b} x}{\sqrt {2}}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{4} \left (\frac {i \sqrt {2} a}{\sqrt {b}}+3\right ),\frac {3}{2},i \sqrt {2} \sqrt {b} x\right )+c_2 L_{\frac {1}{4} \left (-\frac {i \sqrt {2} a}{\sqrt {b}}-3\right )}^{\frac {1}{2}}\left (i \sqrt {2} \sqrt {b} x\right )\right ) \end{align*}

✗ Sympy

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

False