89.4.24 problem 25

Internal problem ID [24346]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:20:50 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} y+2 \left (y^{4}-x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 89

ode:=y(x)+2*(y(x)^4-x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_1^{2}-4 x}+2 c_1}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_1^{2}-4 x}+2 c_1}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_1^{2}-4 x}+2 c_1}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_1^{2}-4 x}+2 c_1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 1.95 (sec). Leaf size: 128

ode=y[x]+2*( y[x]^4-x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {c_1-\sqrt {-4 x+c_1{}^2}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {c_1-\sqrt {-4 x+c_1{}^2}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\sqrt {-4 x+c_1{}^2}+c_1}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt {-4 x+c_1{}^2}+c_1}}{\sqrt {2}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.102 (sec). Leaf size: 102

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

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