23.4.211 problem 211

Internal problem ID [6513]
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 : 211
Date solved : Friday, October 03, 2025 at 02:09:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y y^{\prime \prime }&=-\left (1+y\right ) y^{\prime }+2 x {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 22

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

\begin{align*} y &= 0 \\ y &= c_1 \tanh \left (\frac {\ln \left (x \right )-c_2}{2 c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 11.722 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}}\\ y(x)&\to \frac {1}{2} (\log (x)-c_2) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(8*x**2*y(x)*Derivative(y(x), (x, 2))