75.14.30 problem 356

Internal problem ID [16873]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 356
Date solved : Monday, March 31, 2025 at 03:34:19 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 20

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

\begin{align*} y &= 0 \\ y &= \frac {{\mathrm e}^{c_1 \left (c_2 +x \right )}+1}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.555 (sec). Leaf size: 26

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

\begin{align*} y(x)\to \frac {1+e^{c_1 (x+c_2)}}{c_1} \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) + 1/2 cannot be solved by the factorable group method