23.4.205 problem 205

Internal problem ID [6507]
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 : 205
Date solved : Friday, October 03, 2025 at 02:09:22 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 113

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

\begin{align*} y &= \frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \int x^{a -1} f \left (x \right )d x +x^{-a +1} c_1 -\int f \left (x \right )d x -c_2 \right )}}{a -1} \\ y &= \frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \int x^{a -1} f \left (x \right )d x +x^{-a +1} c_1 -\int f \left (x \right )d x -c_2 \right )}}{-a +1} \\ \end{align*}

✓ Mathematica. Time used: 20.884 (sec). Leaf size: 108

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

\begin{align*} y(x)&\to -\sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2}\\ y(x)&\to \sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) + sqrt(a**2*y(x)**2 - 4*x**2*y(x)