89.31.1 problem 1

Internal problem ID [24946]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 246
Problem number : 1
Date solved : Thursday, October 02, 2025 at 11:36:26 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.134 (sec). Leaf size: 101

ode:=x*diff(y(x),x)^2+y(x)*diff(y(x),x) = 3*y(x)^4; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {3}}{6 \sqrt {-x}} \\ y &= \frac {\sqrt {3}}{6 \sqrt {-x}} \\ y &= 0 \\ y &= -\frac {\sqrt {3}\, \sqrt {x \operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )^{2}}\, \coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )}{6 x} \\ y &= \frac {\sqrt {3}\, \sqrt {x \operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )^{2}}\, \coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_1}{2}\right )}{6 x} \\ \end{align*}

✓ Mathematica. Time used: 0.324 (sec). Leaf size: 94

ode=x*D[y[x],x]^2+y[x]*D[y[x],x]==3*y[x]^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {2 e^{\frac {c_1}{2}}}{-12 x+e^{c_1}}\\ y(x)&\to \frac {2 e^{\frac {c_1}{2}}}{-12 x+e^{c_1}}\\ y(x)&\to 0\\ y(x)&\to -\frac {i}{2 \sqrt {3} \sqrt {x}}\\ y(x)&\to \frac {i}{2 \sqrt {3} \sqrt {x}} \end{align*}

✓ Sympy. Time used: 11.451 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)**2 - 3*y(x)**4 + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt {3}}{6 \sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}, \ y{\left (x \right )} = - \frac {\sqrt {3}}{6 \sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}\right ] \]