23.3.314 problem 316

Internal problem ID [6028]
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 : 316
Date solved : Friday, October 03, 2025 at 01:45:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y+a x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 86

ode:=x^2*(b1*x^2+a1)*y(x)+a*x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {i \sqrt {\operatorname {b1}}\, x^{2}}{2}} \left (\operatorname {KummerM}\left (\frac {a \sqrt {\operatorname {b1}}+i \operatorname {a1} +\sqrt {\operatorname {b1}}}{4 \sqrt {\operatorname {b1}}}, \frac {1}{2}+\frac {a}{2}, i \sqrt {\operatorname {b1}}\, x^{2}\right ) c_1 +\operatorname {KummerU}\left (\frac {a \sqrt {\operatorname {b1}}+i \operatorname {a1} +\sqrt {\operatorname {b1}}}{4 \sqrt {\operatorname {b1}}}, \frac {1}{2}+\frac {a}{2}, i \sqrt {\operatorname {b1}}\, x^{2}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 136

ode=x^2*(a1 + b1*x^2)*y[x] + a*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 2^{\frac {a+1}{4}} x^{\frac {1}{2} (-a-1)} \left (x^2\right )^{\frac {a+1}{4}} e^{\frac {1}{2} i \sqrt {\text {b1}} x^2} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{4} \left (a-\frac {i \text {a1}}{\sqrt {\text {b1}}}+1\right ),\frac {a+1}{2},-i \sqrt {\text {b1}} x^2\right )+c_2 L_{\frac {1}{4} \left (-a+\frac {i \text {a1}}{\sqrt {\text {b1}}}-1\right )}^{\frac {a-1}{2}}\left (-i \sqrt {\text {b1}} x^2\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a1 = symbols("a1") 
b1 = symbols("b1") 
y = Function("y") 
ode = Eq(a*x*Derivative(y(x), x) + x**2*(a1 + b1*x**2)*y(x) + x**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None