23.4.88 problem 88

Internal problem ID [6390]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 88
Date solved : Tuesday, September 30, 2025 at 02:55:00 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} 2 y^{\prime }+a \,x^{2} {y^{\prime }}^{2}+x y^{\prime \prime }&=b \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 74

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

\[ y = \frac {-\sqrt {-a b}\, \int \frac {\operatorname {BesselY}\left (1, \sqrt {-a b}\, x \right ) c_1 +\operatorname {BesselJ}\left (1, \sqrt {-a b}\, x \right )}{x \left (c_1 \operatorname {BesselY}\left (0, \sqrt {-a b}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-a b}\, x \right )\right )}d x +c_2 a}{a} \]

✓ Mathematica. Time used: 60.108 (sec). Leaf size: 116

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

\begin{align*} y(x)&\to \int _1^x\frac {i \sqrt {b} \left (\operatorname {BesselY}\left (1,-i \sqrt {a} \sqrt {b} K[1]\right )-\operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right )}{\sqrt {a} \left (\operatorname {BesselY}\left (0,-i \sqrt {a} \sqrt {b} K[1]\right )+\operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right ) K[1]}dK[1]+c_2 \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded