89.33.1 problem 1

Internal problem ID [24985]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 1
Date solved : Thursday, October 02, 2025 at 11:45:56 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=x {y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 37

ode:=diff(diff(y(x),x),x) = x*diff(y(x),x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \arctan \left (\frac {x}{\sqrt {-x^{2}+c_1}}\right )+c_2 \\ y &= -\arctan \left (\frac {x}{\sqrt {-x^{2}+c_1}}\right )+c_2 \\ \end{align*}

✓ Mathematica. Time used: 19.528 (sec). Leaf size: 57

ode=D[y[x],{x,2}]==x*D[y[x],x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 73.996 (sec). Leaf size: 133

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*Derivative(y(x), x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx, \ y{\left (x \right )} = C_{1} + \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx, \ y{\left (x \right )} = C_{1} - \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx, \ y{\left (x \right )} = C_{1} + \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx, \ y{\left (x \right )} = C_{1} - \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx, \ y{\left (x \right )} = C_{1} + \sqrt {2} \int \sqrt {- \frac {1}{C_{2} + x^{4}}}\, dx\right ] \]