23.4.89 problem 89

Internal problem ID [6391]
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 : 89
Date solved : Tuesday, September 30, 2025 at 02:55:02 PM
CAS classification : [NONE]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(b - x*Derivative(y(x), (x, 2))) + y(