23.3.429 problem 434

Internal problem ID [6143]
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 : 434
Date solved : Tuesday, September 30, 2025 at 02:22:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -3 y+3 x y^{\prime }+\left (2 x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 39

ode:=-3*y(x)+3*x*diff(y(x),x)+(2*x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {LegendreP}\left (\frac {3}{4}, \frac {1}{4}, i \sqrt {2}\, x \right ) c_1 +\operatorname {LegendreQ}\left (\frac {3}{4}, \frac {1}{4}, i \sqrt {2}\, x \right ) c_2 \right ) \left (2 x^{2}+1\right )^{{1}/{8}} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 55

ode=-3*y[x] + 3*x*D[y[x],x] + (1 + 2*x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \sqrt [8]{2 x^2+1} Q_{\frac {3}{4}}^{\frac {1}{4}}\left (i \sqrt {2} x\right )+\frac {i 2^{3/4} c_1 x}{\operatorname {Gamma}\left (\frac {3}{4}\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*Derivative(y(x), x) + (2*x**2 + 1)*Derivative(y(x), (x, 2)) - 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False