89.33.7 problem 7

Internal problem ID [24991]
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 : 7
Date solved : Thursday, October 02, 2025 at 11:46:01 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} 2 a y^{\prime \prime }+{y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 29

ode:=2*a*diff(diff(y(x),x),x)+diff(y(x),x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.228 (sec). Leaf size: 51

ode=2*a*D[y[x],{x,2}]+D[y[x],x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-2 \sqrt {a} \sqrt {x-2 a c_1}\\ y(x)&\to 2 \sqrt {a} \sqrt {x-2 a c_1}+c_2 \end{align*}

✓ Sympy. Time used: 4.651 (sec). Leaf size: 71

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

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