23.4.307 problem 310

Internal problem ID [6609]
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 : 310
Date solved : Friday, October 03, 2025 at 02:09:36 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.069 (sec). Leaf size: 19

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

\[ y = \frac {\operatorname {RootOf}\left (f \left (\textit {\_Z} , c_{1} , c_{2}\right )\right ) x^{2}}{2}+c_{1} x +c_{2} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : 
No algorithms are implemented to solve equation f(_Dummy_37, -_Dummy_37*x + _