82.8.5 problem 36-6

Internal problem ID [21876]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-6
Date solved : Thursday, October 02, 2025 at 08:03:15 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (-1\right )&=0 \\ y^{\prime }\left (-1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.223 (sec). Leaf size: 23

ode:=diff(y(x),x) = x*diff(diff(y(x),x),x)+diff(diff(y(x),x),x)^2; 
ic:=[y(-1) = 0, D(y)(-1) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= x^{2}+4 x +3 \\ y &= -\frac {1}{2} x^{2}+x +\frac {3}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 29

ode=D[y[x],x]==x*D[y[x],{x,2}]+D[y[x],{x,2}]^2; 
ic={y[-1]==0,Derivative[1][y][-1] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x^2}{2}+x+\frac {3}{2}\\ y(x)&\to x^2+4 x+3 \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants