47.4.35 problem 38

Internal problem ID [9809]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 06:42:21 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 0.645 (sec). Leaf size: 14

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

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