23.4.86 problem 86

Internal problem ID [6388]
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 : 86
Date solved : Tuesday, September 30, 2025 at 02:54:58 PM
CAS classification : [[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

\begin{align*} -2 y^{\prime }+2 x {y^{\prime }}^{2}+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 25

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

\[ y = \frac {\ln \left (2\right )}{2}-\frac {\ln \left (3\right )}{2}+\frac {\ln \left (c_1 \,x^{3}+3 c_2 \right )}{2} \]

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to \frac {1}{2} \log \left (2 x^3+3 c_1\right )+c_2 \end{align*}

✓ Sympy. Time used: 0.620 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} + \frac {\log {\left (C_{2} + 2 x^{3} \right )}}{2} \]