23.3.472 problem 478

Internal problem ID [6186]
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 : 478
Date solved : Friday, October 03, 2025 at 01:48:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} c y+b x y^{\prime }+\left (a \,x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 124

ode:=c*y(x)+b*x*diff(y(x),x)+(a*x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (a \,x^{2}+1\right )^{\frac {-b +2 a}{4 a}} \left (\operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {-b +2 a}{2 a}, \sqrt {-a}\, x \right ) c_1 +\operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {-b +2 a}{2 a}, \sqrt {-a}\, x \right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 135

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

\begin{align*} y(x)&\to \left (a x^2+1\right )^{\frac {1}{2}-\frac {b}{4 a}} \left (c_1 P_{\frac {\sqrt {a^2-2 (b+2 c) a+b^2}-a}{2 a}}^{\frac {b}{2 a}-1}\left (i \sqrt {a} x\right )+c_2 Q_{\frac {\sqrt {a^2-2 (b+2 c) a+b^2}-a}{2 a}}^{\frac {b}{2 a}-1}\left (i \sqrt {a} x\right )\right ) \end{align*}

✗ Sympy

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

False