23.4.85 problem 85

Internal problem ID [6387]
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 : 85
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*} x y^{\prime \prime }&=y^{\prime }+x {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.566 (sec). Leaf size: 10

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

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